DETAILED ACTION
A request for continued examination under 37 CFR 1.114, including the fee set forth in 37 CFR 1.17(e), was filed in this application after final rejection. Since this application is eligible for continued examination under 37 CFR 1.114, and the fee set forth in 37 CFR 1.17(e) has been timely paid, the finality of the previous Office action has been withdrawn pursuant to 37 CFR 1.114. Applicant's submission filed on Dec. 22nd, 2025 has been entered.
This action is in response to the amendments filed on Dec. 22nd, 2025. A summary of this action:
Claims 22, 24-29, 31-36, 38-42 have been presented for examination.
Claims 22, 24-29, 31-36, 38-42 are rejected under 35 U.S.C. 112(a) or 35 U.S.C. 112 (pre-AIA ), first paragraph, as failing to comply with the written description requirement
Claims 22, 24-29, 31-36, 38-42 are rejected under 35 U.S.C. 101 because the claimed invention is directed to an abstract idea of both a mathematical concept and mental process without significantly more.
The claims are not rejected in view of the prior art. See the non-final act. in June 2025 for the rationale, which is incorporated herein by reference.
This action is non-final
Notice of Pre-AIA or AIA Status
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .
Response to Arguments/Amendments
Regarding the specification objection
Withdrawn in view of the amendments.
Regarding the § 112(a) rejection and priority
Maintained, updated as necessitated by amendment.
With respect to the remarks, see the rationale in the rejection. The delta operator indicates the difference/change operation, but the variable Ui is what it is taking the difference between (for two values of i, wherein i is a number identifying the “mesh cell i”. To clarify, while ¶ 17 states that the delta “is a difference value for two consecutive iterations”, it is in the context of “where i is an index for an iteration”, whereas in ¶ 30 it is “mesh cell i”, i.e. the variable i has a separate definition in ¶ 30, as such the delta operator, commonly used to indicate a difference (i.e. the applicant is not acting as their own lexicographer in ¶ 17; MPEP § 2111.01(IV.A: “An applicant is entitled to be their own lexicographer and may rebut the presumption that claim terms are to be given their ordinary and customary meaning by clearly setting forth a definition of the term that is different from its ordinary and customary meaning(s) in the specification at the relevant time.”)
To clarify on the plain meaning, see:
Brummer et al., Mathematics: PreCalculus Mathematics at Nebraska. § II for College Algebra, accessed on March 21st, 2026, URL: mathbooks(dot)unl(dot)edu/PreCalculus/Rates-of-Change(dot)html subsection 1, article on Rates of Change. Subsection on “Definition of Slope [of a line]”, see the definition that it is the ratio of change in the y-coordinate over the change in the x-coordinate (i.e. the rise over the run of the line), then see subsection “Notation for Slope: “The symbol Δ (the Greek letter delta) is used in mathematics to denote change in. In particular, Δ y means change in y
-coordinate, and Δ x means change in x-coordinate. We also use the letter m to stand for slope.”
In other words, in ¶ 17,
∆
U
i
-
1
,
w
h
e
r
e
i
n
i is an index for an iteration
a
n
d
"
U
i
s
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h
e
p
o
t
a
t
i
o
n
a
l
s
o
l
u
t
i
o
n
"
Is, in textual form, the difference/change in U the potential solution between U at i-1 iteration and the prior iteration, i.e. U at i-2.
Akin to writing the slope of a line in shorthand compared to long-form in various ways (for the formula y=mx+b):
m
=
S
l
o
p
e
=
r
i
s
e
r
u
n
=
c
h
a
n
g
e
i
n
y
-
c
o
o
r
d
i
n
a
t
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c
h
a
n
g
e
i
n
x
-
c
o
o
r
d
i
n
a
t
e
=
∆
y
∆
x
=
y
2
-
y
1
x
2
-
x
1
Thus, in the context of ¶ 30,
∆
U
i
,
w
h
e
r
e
i
n
i
i
s
a
v
a
l
u
e
r
e
p
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e
s
e
n
t
i
n
g
t
h
e
m
e
s
h
c
e
l
l
= value of potential solution U at mesh cell i minus value of potential solution U at mesh cell i-1.
To summarize, at issue is that math is a highly precise language, and in ¶ 30 the variable i was redefined expressly in ¶ 30 for the purposes of understanding that equation in ¶ 30.
For further clarification, see Mathcentre, Symbols, 2009, accessed via URL: mathcentre(dot)ac(dot)uk/resources/uploaded/mc-bus-symbols-2009-1(dot)pdf, page 2: “The delta notation for the change in a variable The change in the value of a quantity is found by subtracting its initial value from its final value. For example, if the value of an investment is initially £1300 and after some time is found to be £1700, the change in value is 1700 − 1300 = £400. The Greek letter delta (δ, or
∆
) is often used to indicate such a change.”
Regarding the § 101 Rejection
Maintained, and updated as necessitated by amendment.
With respect to the remarks at prong 2, the simulating is math calculations in textual form, not an additional element (see the rejection), and mere data gathering and data outputting is not a practical application. MPEP § 2106.04(d)(I): “The courts have also identified limitations that did not integrate a judicial exception into a practical application: • Merely reciting the words "apply it" (or an equivalent) with the judicial exception, or merely including instructions to implement an abstract idea on a computer, or merely using a computer as a tool to perform an abstract idea, as discussed in MPEP § 2106.05(f); • Adding insignificant extra-solution activity to the judicial exception, as discussed in MPEP § 2106.05(g); and • Generally linking the use of a judicial exception to a particular technological environment or field of use, as discussed in MPEP § 2106.05(h)”, e.g. see the additional elements in example 45, claim 1 and 3; example 46, claim 1.
Also, to clarify see MPEP § 2106 (II)(A)(2): “See, e.g., RecogniCorp, LLC v. Nintendo Co., 855 F.3d 1322, 1327, 122 USPQ2d 1377 (Fed. Cir. 2017) ("Adding one abstract idea (math) to another abstract idea (encoding and decoding) does not render the claim non-abstract"); Genetic Techs. Ltd. v. Merial LLC, 818 F.3d 1369, 1376, 118 USPQ2d 1541, 1546 (Fed. Cir. 2016) (eligibility "cannot be furnished by the unpatentable law of nature (or natural phenomenon or abstract idea) itself."). For a claim reciting a judicial exception to be eligible, the additional elements (if any) in the claim must "transform the nature of the claim" into a patent-eligible application of the judicial exception, Alice Corp., 573 U.S. at 217, 110 USPQ2d at 1981, either at Prong Two or in Step 2B.” and MPEP § 2106.04(d) to further clarify.
With respect to the 2B remarks, these merely allege the abstract idea itself provides/furnishes the improvement - see MPEP § 2106.05(I): “An inventive concept "cannot be furnished by the unpatentable law of nature (or natural phenomenon or abstract idea) itself." Genetic Techs. Ltd. v. Merial LLC, 818 F.3d 1369, 1376, 118 USPQ2d 1541, 1546 (Fed. Cir. 2016). See also Alice Corp., 573 U.S. at 21-18, 110 USPQ2d at 1981 (citing Mayo, 566 U.S. at 78, 101 USPQ2d at 1968 (after determining that a claim is directed to a judicial exception, "we then ask, ‘[w]hat else is there in the claims before us?") (emphasis added)); RecogniCorp, LLC v. Nintendo Co., 855 F.3d 1322, 1327, 122 USPQ2d 1377 (Fed. Cir. 2017) ("Adding one abstract idea (math) to another abstract idea (encoding and decoding) does not render the claim non-abstract"). Instead, an "inventive concept" is furnished by an element or combination of elements that is recited in the claim in addition to (beyond) the judicial exception, and is sufficient to ensure that the claim as a whole amounts to significantly more than the judicial exception itself. Alice Corp., 573 U.S. at 27-18, 110 USPQ2d at 1981 (citing Mayo, 566 U.S. at 72-73, 101 USPQ2d at 1966).” – see MPEP § 2106.05(a): “It is important to note, the judicial exception alone cannot provide the improvement.”
Regarding the remarks about § 103
There was no § 103 rejection in the prior action to respond to in view of the prior amendments (see the June 2025 non-final act.), as such these are moot.
Priority
Applicant’s claim for the benefit of a prior-filed application under 35 U.S.C. 119(e) or under 35 U.S.C. 120, 121, 365(c), or 386(c) is acknowledged. Applicant has not complied with one or more conditions for receiving the benefit of an earlier filing date under 35 U.S.C. 119(e) and 120 as follows:
The later-filed application must be an application for a patent for an invention which is also disclosed in the prior application (the parent or original nonprovisional application or provisional application). The disclosure of the invention in the parent application and in the later-filed application must be sufficient to comply with the requirements of 35 U.S.C. 112(a) or the first paragraph of pre-AIA 35 U.S.C. 112, except for the best mode requirement. See Transco Products, Inc. v. Performance Contracting, Inc., 38 F.3d 551, 32 USPQ2d 1077 (Fed. Cir. 1994).
The disclosure of the prior-filed application, Application No. 62/933,213, fails to provide adequate support or enablement in the manner provided by 35 U.S.C. 112(a) or pre-AIA 35 U.S.C. 112, first paragraph for one or more claims of this application. The disclosure of the prior-filed application, Application No. 16/705,921, fails to provide adequate support or enablement in the manner provided by 35 U.S.C. 112(a) or pre-AIA 35 U.S.C. 112, first paragraph for one or more claims of this application.
The independent claims (using claim 22 as representative) recite: determining if the new potential first solution satisfies a convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential first solution of consecutive iterations and values of the new potential first solution, wherein the first iterative process stops when the convergence criterion is satisfied by the new potential first solution;
determining if the new potential second solution satisfies he convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential second solution of consecutive iterations and values of the new potential second solution, wherein the second iterative process is stopped when the convergence criterion is satisfied by the new potential second solution;
See ¶ 30. What is claimed is not sufficiently described.
The ‘123 and ‘921 share the same disclosure. In particular, the Examiner notes that ¶ 30 provides a definition for the top-most part of the equation in ¶ 30, i.e. the bottom portion of the equation is the 1-norm of Ui; but the top portion must use the equation defined.
At issue is that “i” this definition is an integer identifying the “mesh cell i” – it is no longer an integer representing the “ith iteration” (¶ 27, and elsewhere in the instant specification). Thus, the equation in ¶ 30 has no “absolute difference between consecutive iterations” because
∆
U
i
as read in combination with “i” being the mesh cell integer is no longer indicating a difference between consecutive iterations, as “i” is expressly re-defined in ¶ 30 to have a distinct meaning from the “i” variable used elsewhere in the instant disclosure.
The dependent claims inherit this deficiency.
Claim Rejections - 35 USC § 112(a)
The following is a quotation of the first paragraph of 35 U.S.C. 112(a):
(a) IN GENERAL.—The specification shall contain a written description of the invention, and of the manner and process of making and using it, in such full, clear, concise, and exact terms as to enable any person skilled in the art to which it pertains, or with which it is most nearly connected, to make and use the same, and shall set forth the best mode contemplated by the inventor or joint inventor of carrying out the invention.
The following is a quotation of the first paragraph of pre-AIA 35 U.S.C. 112:
The specification shall contain a written description of the invention, and of the manner and process of making and using it, in such full, clear, concise, and exact terms as to enable any person skilled in the art to which it pertains, or with which it is most nearly connected, to make and use the same, and shall set forth the best mode contemplated by the inventor of carrying out his invention.
Claims 22, 24-29, 31-36, 38-42 are rejected under 35 U.S.C. 112(a) or 35 U.S.C. 112 (pre-AIA ), first paragraph, as failing to comply with the written description requirement. The claim(s) contains subject matter which was not described in the specification in such a way as to reasonably convey to one skilled in the relevant art that the inventor or a joint inventor, or for applications subject to pre-AIA 35 U.S.C. 112, the inventor(s), at the time the application was filed, had possession of the claimed invention. The dependent claims inherent the deficiencies of the claims they depend upon.
See MPEP § 2163(II)(A): “For example, in Hyatt v. Dudas, 492 F.3d 1365, 1371, 83 USPQ2d 1373, 1376-1377 (Fed. Cir. 2007), the examiner made a prima facie case by clearly and specifically explaining why applicant’s specification did not support the particular claimed combination of elements, even though applicant’s specification listed each and every element in the claimed combination. The court found the "examiner was explicit that while each element may be individually described in the specification, the deficiency was lack of adequate description of their combination" and, thus, "[t]he burden was then properly shifted to [inventor] to cite to the examiner where adequate written description could be found or to make an amendment to address the deficiency." Id.;” – also, see MPEP § 2163(I) for Lockwood v. American Airlines.
The independent claims (using claim 22 as representative, other claims have similar limitations rejected under a similar rationale) recite: determining if the new potential first solution satisfies a convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential first solution of consecutive iterations and values of the new potential first solution, wherein the first iterative process stops when the convergence criterion is satisfied by the new potential first solution;
determining if the new potential second solution satisfies he convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential second solution of consecutive iterations and values of the new potential second solution, wherein the second iterative process is stopped when the convergence criterion is satisfied by the new potential second solution;
See ¶ 30. What is claimed is not sufficiently described.
In particular, the Examiner notes that ¶ 30 provides a definition for the top-most part of the equation in ¶ 30, i.e. the bottom portion of the equation is the 1-norm of Ui; but the top portion must use the equation defined.
At issue is that “i” this definition is an integer identifying the “mesh cell i” – it is no longer an integer representing the “ith iteration” (¶ 27, and elsewhere in the instant specification). Thus, the equation in ¶ 30 has no “absolute difference between consecutive iterations…” because
∆
U
i
as read in combination with “i” being the mesh cell integer is no longer indicating a difference between consecutive iterations, as “i” is expressly re-defined in ¶ 30 to have a distinct meaning from the “i” variable used elsewhere in the instant disclosure.
Claim Rejections - 35 USC § 101
35 U.S.C. 101 reads as follows:
Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent therefor, subject to the conditions and requirements of this title.
Claims 22, 24-29, 31-36, 38-42 are rejected under 35 U.S.C. 101 because the claimed invention is directed to an abstract idea of both a mathematical concept and mental process without significantly more.
Step 1
Claim 22 is directed towards the statutory category of a process.
Claim 29 is directed towards the statutory category of an apparatus.
Claim 36 is directed towards the statutory category of an article of manufacture.
Claims 29 and 36, and the dependents thereof, are rejected under a similar rationale as representative claim 22, and the dependents thereof.
Step 2A – Prong 1
The claims recite an abstract idea of both a mental process and mathematical concept.
To describe the abstract idea recited in the present claims at a high level of abstraction, the claims recite the abstract idea of determining when to stop performing iterative mathematical calculations based on using another mathematical calculation which is the “combination of a sum of absolute values and a sum of absolute differences between consecutive iterations of the one or more previously calculated potential … solutions and the new potential … solution” and comparing the result of this calculation to a “mesh size independent tolerance value”, so as to determine convergence in the iterative math calculations (i.e. determining when the results of the iterative math calculations start approximately repeating/are close to each other for a subset of the iterations), which is an abstract idea of both a mental process and a mathematical concept as discussed in detail below with respect to the ordered combination of claimed features.
See MPEP § 2106.04: “...In other claims, multiple abstract ideas, which may fall in the same or different groupings, or multiple laws of nature may be recited. In these cases, examiners should not parse the claim. For example, in a claim that includes a series of steps that recite mental steps as well as a mathematical calculation, an examiner should identify the claim as reciting both a mental process and a mathematical concept for Step 2A Prong One to make the analysis clear on the record.”
To clarify, see the USPTO 101 training examples, available at https://www.uspto.gov/patents/laws/examination-policy/subject-matter-eligibility.
The mathematical concept recited in claim 22 is:
… simulating a characteristic of the physical system based on multiple models of different mesh sizes for modeling the physical system by iterative processes calculating potential solutions for a set of equations associated with the characteristic of the physical system the multiple models including a first model of a first mesh size and a second model of a second mesh size; - math calculations in textual form. See ¶ 2 of the instant disclosure: “…A mesh includes elements that are typically simple shapes such as tetrahedrons, hexahedrons, prisms, and other volumetric (three dimensional - 3D) shapes. These simple shapes have faces, edges, and vertices. A mesh includes a collection of these elements, which can be referred to as mesh elements, and a mathematical description of the relationship between the nodes, edges, and faces of the mesh elements. In a numerical analysis of physical systems, complicated three dimensional shapes may be broken down to (i.e., approximated by) a mesh representation. The mesh representation may subsequently be used in setting up and solving equations for analysis using a discretization method (e.g. finite volume analysis) which may include numerical solution of partial differential equations based on boundary conditions”, then see ¶ 17 including: “Referring to Figure 2, system 200 is a simulation process which solves a system of equations to obtain an increment …to the potential solution … in an iterative process, where i is an index for an iteration, U is the potential solution,/),, is a difference value for two consecutive iterations, and F represents one or more equations characterizing a physical system. The system of equations is set up with one or more boundary conditions to simulate a characteristic of the physical system that is modeled by a number of mesh elements…”, also see ¶ 22: “Solver module 158 can solve for a potential solution based on a set of equations, or physics 154, and a mesh model from mesh generator 156 generating solver results 160. Here the set of equations describe one or more characteristics being simulated.”, ¶ 23: “Potential solution calculation module 301 can calculate one or more potential solutions based on a set of equations”, ¶ 25: “To configure a simulation, a formulation ( or a set of equations or an objective function) is applied to a model of the physical system. The formulation specifies the properties of the model and/or the types of simulations to simulate…”, then see ¶¶ 26-27
To clarify on the “mesh” and “different mesh sizes”, these are considered mathematical relationships in geometry for use in the math calculation, as the mesh is a series of mathematical relationships representing the geometry of an shape by representing small portions of the geometry (i.e. with smaller shapes), and the mathematical relationships between these smaller shapes, e.g. representing the geometry of a square by using four smaller squares (example of mesh element), wherein the four smaller squares have their dimensions and locations related to the larger square by math relationships (e.g., suppose the square is of area 1, then each smaller square would have an area of ¼ (i.e. the math relationship of the area of the larger square, divided by the number of smaller squares), wherein the positioning of the smaller squares is mathematically described by the relationship between the smaller squares and the larger square, e.g. suppose the origin of the larger square is a x=0,y=0; with an opposing corner at x=1, y=1 (x-y axes), then the four smaller squares would be mathematically located to have their inside corner at the center of the larger square, i.e. x= ½, y=½. Hence, ¶ 2: “A mesh includes a collection of these elements, which can be referred to as mesh elements, and a mathematical description of the relationship between the nodes [e.g. corners], edges, and faces of the mesh elements”. The recitation of “different mesh sizes” is considered a geometrical relationship between the size of the mesh elements, and the size of the object to be represented, e.g. if the above larger square was to be represented by 16 smaller squares, then the smaller squares would have a size, e.g. area, of 1/16, by the math relationship of the area of the larger square to the number of smaller squares. See ¶ 2 including: “In a numerical analysis of physical systems, complicated three dimensional shapes may be broken down to (i.e., approximated by) a mesh representation” and then see ¶ 33 to further clarify: “Referring to Figures 6A-6C, the respective mesh sizes are 320 x 64, 640 x 128, and 1280 x 256 [these are the number of mesh elements to represent the geometry, and note that each subsequent mesh size is 2x the previous size, e.g. by subdividing each prior element into two elements, or by remeshing to achieve a similar effect]”
A visual example of such geometrical mathematical relationships of an example of a mesh is provided below:
PNG
media_image1.png
751
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Greyscale
To further clarify, the Examiner notes that mathematical relationships in geometry are discussed in MPEP § 2106.04(a)(2) “iii. a mathematical relationship between enhanced directional radio activity and antenna conductor arrangement (i.e., the length of the conductors with respect to the operating wave length and the angle between the conductors), Mackay Radio & Tel. Co. v. Radio Corp. of America, 306 U.S. 86, 91, 40 USPQ 199, 201 (1939) (while the litigated claims 15 and 16 of U.S. Patent No. 1,974,387 expressed this mathematical relationship using a formula that described the angle between the conductors, other claims in the patent (e.g., claim 1) expressed the mathematical relationship in words)”
calculating a new potential first solution for the set of equations using the first model for a first iterative process, the first iterative process having one or more previously calculated potential first solutions for the set of equations using the first model; calculating a new potential second solution for the set of equations using the second model for a second iterative process, the second iterative process having one or more previously calculated potential second solutions for the set of equations using the second model; - math calculations in textual form. To clarify on the BRI, see the instant disclosure, ¶¶ 2, 17, 22-23, 25-27 as discussed above.
determining if the new potential first solution satisfies a convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential first solution of consecutive iterations and values of the new potential first solution, wherein the first iterative process stops when the convergence criterion is satisfied by the new potential first solution; determining if the new potential second solution satisfies he convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential second solution of consecutive iterations and values of the new potential second solution, wherein the second iterative process is stopped when the convergence criterion is satisfied by the new potential second solution; - math calculations/equations/relationships in textual form. To clarify on the BRI, see the instant disclosure, ¶¶ 2, 17, 22-23, 25-27 as discussed above, then see ¶¶ 18-19 and ¶¶ 27-30, including the equation in ¶¶ 28 and 30 which gives an example of how to use a mathematical equation/relationship to perform the comparison.
To clarify on the above, see ¶¶ 17-20 including: “ … The iterative process is stopped when a potential solution which satisfies a convergence condition is found, e.g., if the potential
solution …matches a target (true) solution … A typical approach is to use a residual … [see the equation]… However, a residual approach requires the convergence tolerance… to be adjusted
based on the value o…” and ¶ 32: “Here, the simulation problem has no steady state solution and a simulation based on a residual value would not result in convergence. However, simulating the aerodynamics of the truck using a real-time simulation environment 150 (having the mesh-size independent stop criterion) of Figure 3A, can achieve a stop condition.” – i.e. the instant disclosure describes that the problem to be solved is a mathematical problem in the use of the mathematical calculation of a “residual” to determine the “closeness of the potential solution to a target solution” (¶ 18) such as to determine a “stop condition”/”convergence” when performing iterative calculations (i.e. when the results of each iterative calculation become close to each other) because the residual calculation is “mesh size dependent.” (¶ 20), and the instant disclosure then describes a mathematical solution, in both textual form and mathematical form (¶¶ 28-30), to this mathematical problem. This is a mathematical solution to a mathematical problem by using different mathematical equations/relationships/calculations (¶¶ 28-30) instead of the residual (¶ 18) to solve the math problem.
While the Examiner finds that the plain meaning of the terms used in the present claims is clear from the instant disclosure (MPEP § 2111.01(III)) and consistent with how they are used in the prior art, should further clarification be needed to the plain meaning, see Motamed, “Newton’s Method”, Course Notes for Math/CS 471 High Performance Scientific Computing, University of New Mexico, Fall 2020, URL: math(dot)unm(dot)edu/%7Emotamed/Teaching/OLD/Fall20/HPSC/newton(dot)html – section “Iterative Methods”, for “Definition. An iterative method is a mathematical procedure that generates a sequence of improving approximate solutions to a problem. An iterative method starts with an initial approximation, which is often called an initial guess” which is a definition consistent with both the instant disclosure and the instant claims for its description of “…iterative process calculation potential solution…” as discussed above, then see remaining parts of “Iterative Methods”, including “Stopping criterion I”: “A stopping criterion tells us when to terminate an iterative algorithm…This will determine the number of iterations n necessary to obtain an approximate solution xn within the error tolerance…”.
Also see the International Association for the Engineering Modelling, Analysis, Simulation Community (NAFEMS), Webpage “What Is Convergence?”, accessed July 26th, 2024, URL: www(dot)nafems(dot)org/publications/guidelines-for-good-convergence-in-cfd/what-is-convergence/, see “one definition is ‘More formally, in mathematics, convergence describes limiting behaviour, particularly of an infinite sequence or series toward some limit . To assert convergence is to claim the existence of a limit, which may be itself unknown. For any fixed standard of accuracy, you can always be sure to be within it, provided you have gone far enough.’” And also see “As this definition indicates, the exact solution to the iterative problem is unknown, but you want to be sufficiently close to the solution for a particular required level of accuracy…Convergence is also often measured by the level of residuals, the amount by which discretized equations are not satisfied, and not by the error in the solution.”
Also see Engeln-Mullges, “Numerical Algorithms With C”, 1996, page VI: “The book is not meant to be a textbook on 'Numerical Analysis': we include no single proof, hardly any examples and no problems. Our emphasis when writing this book lay in explaining the principles of various methods, and in describing efficient and proven algorithms for the standard methods of Numerical Analysis, as well as documenting and developing them…”, see page VII including ¶¶ 2-3, then see § 2.3.1 including page 15: “If the sequence {xCv)} converges, i.e., if [an equation for a mathematical definition of convergence]”, also see page 16 ¶ 1, then see § 2.3.3 “Convergence and Error Estimates of Iterative Procedures” including page 19.
Ansys, article on “What is Finite Element Analysis (FEA)?”, accessed June 9th, 2025, URL: ansys(dot)com/simulation-topics/what-is-finite-element-analysis: “Finite element analysis (FEA) is the process of predicting an object’s behavior based on calculations made with the finite element method (FEM). While FEM is a mathematical technique, FEA is the interpretation of the results FEM provides…. FEM uses math to break complex systems into smaller, simpler pieces, or “elements.” It then applies differential equations to each element individually, using the power of computers to divide, then conquer engineering problems… FEA is the application of FEM equations and is the basis of many types of simulation software. It’s used to validate and test designs safely, quickly, and economically by creating virtual models of real-world assets…. Finite Element Analysis works by discretizing the domain of interest and then assembling physics equations to solve the engineering problem at hand… Process: Divide the object into finite elements via meshing and apply the relevant physics representations and/or equations to each element. Then assemble the equations and solve them.”
Also see MPEP § 2106.04(I): “The Supreme Court’s decisions make it clear that judicial exceptions need not be old or long-prevalent, and that even newly discovered or novel judicial exceptions are still exceptions. For example, the mathematical formula in Flook, the laws of nature in Mayo, and the isolated DNA in Myriad were all novel or newly discovered, but nonetheless were considered by the Supreme Court to be judicial exceptions because they were "‘basic tools of scientific and technological work’ that lie beyond the domain of patent protection." Myriad, 569 U.S. 576, 589, 106 USPQ2d at 1976, 1978 (noting that Myriad discovered the BRCA1 and BRCA1 genes and quoting Mayo, 566 U.S. 71, 101 USPQ2d at 1965); Flook, 437 U.S. at 591-92, 198 USPQ2d at 198 ("the novelty of the mathematical algorithm is not a determining factor at all"); Mayo, 566 U.S. 73-74, 78, 101 USPQ2d 1966, 1968 (noting that the claims embody the researcher's discoveries of laws of nature). The Supreme Court’s cited rationale for considering even "just discovered" judicial exceptions as exceptions stems from the concern that "without this exception, there would be considerable danger that the grant of patents would ‘tie up’ the use of such tools and thereby ‘inhibit future innovation premised upon them.’" Myriad, 569 U.S. at 589, 106 USPQ2d at 1978-79 (quoting Mayo, 566 U.S. at 86, 101 USPQ2d at 1971). See also Myriad, 569 U.S. at 591, 106 USPQ2d at 1979 ("Groundbreaking, innovative, or even brilliant discovery does not by itself satisfy the §101 inquiry."). The Federal Circuit has also applied this principle, for example, when holding a concept of using advertising as an exchange or currency to be an abstract idea, despite the patentee’s arguments that the concept was "new". Ultramercial, Inc. v. Hulu, LLC, 772 F.3d 709, 714-15, 112 USPQ2d 1750, 1753-54 (Fed. Cir. 2014). Cf. Synopsys, Inc. v. Mentor Graphics Corp., 839 F.3d 1138, 1151, 120 USPQ2d 1473, 1483 (Fed. Cir. 2016) ("a new abstract idea is still an abstract idea") (emphasis in original).” And MPEP § 2106.05(a): “It is important to note, the judicial exception alone cannot provide the improvement.”
Under the broadest reasonable interpretation, the claim recites a mathematical concept – the above limitations are steps in a mathematical concept such as mathematical relationships, mathematical formulas or equations, and mathematical calculations. If a claim, under its broadest reasonable interpretation, is directed towards a mathematical concept, then it falls within the Mathematical Concepts grouping of abstract ideas. In addition, as per MPEP § 2106.04(a)(2): “It is important to note that a mathematical concept need not be expressed in mathematical symbols, because "[w]ords used in a claim operating on data to solve a problem can serve the same purpose as a formula." In re Grams, 888 F.2d 835, 837 and n.1, 12 USPQ2d 1824, 1826 and n.1 (Fed. Cir. 1989). See, e.g., SAP America, Inc. v. InvestPic, LLC, 898 F.3d 1161, 1163, 127 USPQ2d 1597, 1599 (Fed. Cir. 2018)”
See MPEP § 2106.04(a)(2).
To clarify, see the USPTO 101 training examples, available at https://www.uspto.gov/patents/laws/examination-policy/subject-matter-eligibility. In particular examples 47 and 49, for their analysis at step 2A, prong 1.
The mental process recited in claim 22 is:
calculating a new potential first solution for the set of equations using the first model for a first iterative process, the first iterative process having one or more previously calculated potential first solutions for the set of equations using the first model; calculating a new potential second solution for the set of equations using the second model for a second iterative process, the second iterative process having one or more previously calculated potential second solutions for the set of equations using the second model; - a mental process of a series of mental evaluations, such as by an engineer or a math professor, using pen and paper, and/or a whiteboard and marker (when the professor is teaching a class), and/or a calculator, or other physical aids, to perform a series of simple calculations, such as by using Newton’s method to find an approximate solution to a math equation (example of a model) by performing a series of simple calculations. An illustrative example of Newton’s method is provided below to clarify on how a person would be able to perform iterative calculations. Other calculations may be used as well for the mental process, as the claim recites with no particularity what equation is being used for the calculation
determining if the new potential first solution satisfies a convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential first solution of consecutive iterations and values of the new potential first solution, wherein the first iterative process stops when the convergence criterion is satisfied by the new potential first solution; determining if the new potential second solution satisfies he convergence criterion based on a comparison between the mesh size independent tolerance value and a combination of a sum of absolute values of the new potential first solution modified by corresponding cell sizes and a sum of absolute differences between values of the one or more previously calculated potential second solution of consecutive iterations and values of the new potential second solution, wherein the second iterative process is stopped when the convergence criterion is satisfied by the new potential second solution; - a mental process of a mental judgement based on mental evaluations, such as aided by physical aids like pen, paper, and/or a calculator, as discussed above. For example, a person having performed the calculations above using Newton’s method needs to then judge whether they converge, so they may readily make a mental evaluation, such as by a calculation of this equation (¶ 30 of the specification) wherein a calculator or a slide rule, with pen and paper, may be used as a physical aid in performing this calculation, and then mentally compare, in a mental judgement, the value resultant from the mental evaluation of the calculation to another value so as to judge whether the iterative calculations converged.
To give an example a mental process of an iterative method of calculation, see Australian Mathematical Sciences Institute (AMSI), webpage on “Newton's method”, accessed July 23rd, 2024, URL: amsi(dot)org(dot)au/ESA_Senior_Years/SeniorTopic3/3j/3j_2content_2(dot)html – see the section “Finding a solution with geometry”, then see the “Example” reproduced below which shows a series of simple iterative calculations that converge “to 11 decimal places” “after 4 iterations”, and see the remaining examples which provide additional applications of Newton’s method to other math problems.
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Furthermore, the Examiner also notes that iterative calculations were performed before the advent of the computer. See Encyclopedia Britannica, Article on “Newton’s method”, accessed July 26th, 2024, URL: www(dot)britannica(dot)com/science/Newtons-method, “The method uses successive approximations to find a value of x that best gives a value of zero in the polynomial expression. The method was devised by Isaac Newton in 1669 on the basis of a method of French mathematician Francois Viète (who may in turn have learned of it from the work of Persian astronomer al-Kāshī). In 1690 English mathematician Joseph Raphson greatly simplified Newton’s method to its current form.”, then see the “Method”, and the “Example”.
Under the broadest reasonable interpretation, these limitations are process steps that cover mental processes including an observation, evaluation, judgment or opinion that could be performed in the human mind or with the aid of physical aids but for the recitation of a generic computer component. If a claim, under its broadest reasonable interpretation, covers a mental process but for the recitation of generic computer components, then it falls within the "Mental Process" grouping of abstract ideas. A person would readily be able to perform this process either mentally or with the assistance of physical aids. See MPEP § 2106.04(a)(2).
To clarify, see the USPTO 101 training examples, available at https://www.uspto.gov/patents/laws/examination-policy/subject-matter-eligibility. In particular, with respect to the physical aids, see example # 45, analysis of claim 1 under step 2A prong 1, including: “Note that even if most humans would use a physical aid (e.g., pen and paper, a slide rule, or a calculator) to help them complete the recited calculation, the use of such physical aid does not negate the mental nature of this limitation.”; also see example # 49, analysis of claim 1, under step 2A prong 1: “Moreover, the recited mathematical calculation is simple enough that it can be practically performed in the human mind. Even if most humans would use a physical aid, like a pen and paper or a calculator, to make such calculations, the use of a physical aid would not negate the mental nature of this limitation.”
As such, the claims recite an abstract idea of both a mental process and mathematical concept.
Step 2A, prong 2
The claimed invention does not recite any additional elements that integrate the judicial exception into a practical application. Refer to MPEP §2106.04(d).
The following limitations are merely reciting the words "apply it" (or an equivalent) with the judicial exception, or merely including instructions to implement an abstract idea on a computer, or merely using a computer as a tool to perform an abstract idea, as discussed in MPEP § 2106.05(f), including the “Use of a computer or other machinery in its ordinary capacity for economic or other tasks (e.g., to receive, store, or transmit data) or simply adding a general purpose computer or computer components after the fact to an abstract idea (e.g., a fundamental economic practice or mathematical equation) does not integrate a judicial exception into a practical application or provide significantly more”:
Claim 22 - A computer-implemented …; claim 29 - A system, comprising: a network interface for connecting to a network; a memory storing instructions; one or more processors coupled to the network interface and the memory, the one or more processors executing the instructions from the memory, the instructions including; claim 36 - A non-transitory computer-readable medium storing instructions for commanding one or more processors to perform a method, the method comprising:
The following limitations are adding insignificant extra-solution activity to the judicial exception, as discussed in MPEP § 2106.05(g):
accessing a mesh size independent tolerance value of a configuration input; - mere data gathering
d generating an output to indicate the simulated characteristic of the physical system in response to the new potential first solution and the new potential second solution satisfying the convergence criterion based on the mesh size independent tolerance value. – mere data outputting
In addition, should it be found that the “simulating…” limitation as a whole is not part of the abstract idea, then the Examiner submits that this would be considered as a insignificant extra-solution activity of mere data gathering of data for use in the later recited “calculating…” steps. In addition, the Examiner notes that this is also considered as generally linking to a particular field of use wherein iterative calculations are used for “simulating a characteristic of [a] physical system…”, as there are many other fields of use that use iterative calculations wherein a convergence check is performed (e.g. Newton’s method to solve an arbitrary polynomial equation as discussed above with respect to step 2A prong 1), and this is also considered as part of the mere instructions to apply a computer to implement the abstract idea, when this limitation is read in view of ¶ 2: “The simulation of physical systems often employs a mesh which is used in a discretization method ( e.g. finite volume method) to provide outputs for the simulation… The mesh representation may subsequently be used in setting up and solving equations for analysis using a discretization method (e.g. finite volume analysis) which may include numerical solution of partial differential equations based on boundary conditions. Other analysis methods can be used in other examples. Mesh-based analysis techniques are used widely in the fields of computational fluid dynamics (CFD), aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and other fields of engineering as well as manufacturing and fabrication processes, such as additive manufacturing processes” and ¶ 22 of the instant disclosure: “Solver module 158 can be any type of solver, such as a finite element solver, having a mesh-size independent stopping criterion. Solver module 158 can solve for a potential solution based on a set of equations, or physics 154, and a mesh model from mesh generator 156 generating solver results 160. Here the set of equations describe one or more characteristics being simulated.”
A claim that integrates a judicial exception into a practical application will apply, rely on, or use the judicial exception in a manner that imposes a meaningful limit on the judicial exception, such that the claim is more than a drafting effort designed to monopolize the judicial exception. See MPEP § 2106.04(d).
The claimed invention does not recite any additional elements that integrate the judicial exception into a practical application. Refer to MPEP §2106.04(d).
Step 2B
The claimed invention does not recite any additional elements/limitations that amount to significantly more.
The following limitations are merely reciting the words "apply it" (or an equivalent) with the judicial exception, or merely including instructions to implement an abstract idea on a computer, or merely using a computer as a tool to perform an abstract idea, as discussed in MPEP § 2106.05(f), including the “Use of a computer or other machinery in its ordinary capacity for economic or other tasks (e.g., to receive, store, or transmit data) or simply adding a general purpose computer or computer components after the fact to an abstract idea (e.g., a fundamental economic practice or mathematical equation) does not integrate a judicial exception into a practical application or provide significantly more”:
Claim 22 - A computer-implemented …; claim 29 - A system, comprising: a network interface for connecting to a network; a memory storing instructions; one or more processors coupled to the network interface and the memory, the one or more processors executing the instructions from the memory, the instructions including; claim 36 - A non-transitory computer-readable medium storing instructions for commanding one or more processors to perform a method, the method comprising:
The following limitations are adding insignificant extra-solution activity to the judicial exception, as discussed in MPEP § 2106.05(g):
accessing a mesh size independent tolerance value of a configuration input; - mere data gathering
d generating an output to indicate the simulated characteristic of the physical system in response to the new potential first solution and the new potential second solution satisfying the convergence criterion based on the mesh size independent tolerance value. – mere data outputting
In addition, should it be found that the “simulating…” limitation as a whole is not part of the abstract idea, then the Examiner submits that this would be considered as a insignificant extra-solution activity of mere data gathering of data for use in the later recited “calculating…” steps. In addition, the Examiner notes that this is also considered as generally linking to a particular field of use wherein iterative calculations are used for “simulating a characteristic of [a] physical system…”, as there are many other fields of use that use iterative calculations wherein a convergence check is performed (e.g. Newton’s method to solve an arbitrary polynomial equation as discussed above with respect to step 2A prong 1), and this is also considered as part of the mere instructions to apply a computer to implement the abstract idea, when this limitation is read in view of ¶ 2: “The simulation of physical systems often employs a mesh which is used in a discretization method ( e.g. finite volume method) to provide outputs for the simulation… The mesh representation may subsequently be used in setting up and solving equations for analysis using a discretization method (e.g. finite volume analysis) which may include numerical solution of partial differential equations based on boundary conditions. Other analysis methods can be used in other examples. Mesh-based analysis techniques are used widely in the fields of computational fluid dynamics (CFD), aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and other fields of engineering as well as manufacturing and fabrication processes, such as additive manufacturing processes” and ¶ 22 of the instant disclosure: “Solver module 158 can be any type of solver, such as a finite element solver, having a mesh-size independent stopping criterion. Solver module 158 can solve for a potential solution based on a set of equations, or physics 154, and a mesh model from mesh generator 156 generating solver results 160. Here the set of equations describe one or more characteristics being simulated.”
In addition, the above insignificant extra-solution activities are also considered as well-understood, routine, and conventional activities, as discussed in MPEP § 2106.05(d):
accessing a mesh size independent tolerance value of a configuration input; - this is considered similar to the example WURC activity as discussed in MPEP § 2106.05(d)(II) of: “iii. Electronic recordkeeping, Alice Corp. Pty. Ltd. v. CLS Bank Int'l, 573 U.S. 208, 225, 110 USPQ2d 1984 (2014) (creating and maintaining "shadow accounts"); Ultramercial, 772 F.3d at 716, 112 USPQ2d at 1755 (updating an activity log); iv. Storing and retrieving information in memory, Versata Dev. Group, Inc. v. SAP Am., Inc., 793 F.3d 1306, 1334, 115 USPQ2d 1681, 1701 (Fed. Cir. 2015); OIP Techs., 788 F.3d at 1363, 115 USPQ2d at 1092-93;”
and generating an output to indicate the simulated characteristic of the physical system in response to the new potential first solution and the new potential second solution satisfying the convergence criterion based on the mesh size independent tolerance value - this is considered similar to the example WURC activity as discussed in MPEP § 2106.05(d)(II) of: “iii. Electronic recordkeeping, Alice Corp. Pty. Ltd. v. CLS Bank Int'l, 573 U.S. 208, 225, 110 USPQ2d 1984 (2014) (creating and maintaining "shadow accounts"); Ultramercial, 772 F.3d at 716, 112 USPQ2d at 1755 (updating an activity log); iv. Storing and retrieving information in memory, Versata Dev. Group, Inc. v. SAP Am., Inc., 793 F.3d 1306, 1334, 115 USPQ2d 1681, 1701 (Fed. Cir. 2015); OIP Techs., 788 F.3d at 1363, 115 USPQ2d at 1092-93;… iv. Presenting offers and gathering statistics, OIP Techs., 788 F.3d at 1362-63, 115 USPQ2d at 1092-93; v. Determining an estimated outcome and setting a price, OIP Techs., 788 F.3d at 1362-63, 115 USPQ2d at 1092-93; and” – also see ¶¶ 1-2 and 17-19 of the instant disclosure for additional evidence. Should more evidence be required, see the below citations for the simulating step.
In addition, should it be found that the “simulating…” limitation as a whole is not part of the abstract idea, then the Examiner submits that this would be considered as a insignificant extra-solution activity of mere data gathering of data for use in the later recited “calculating…” steps. In addition, the Examiner notes that this is also considered as generally linking to a particular field of use wherein iterative calculations are used for “simulating a characteristic of [a] physical system…”, as there are many other fields of use that use iterative calculations wherein a convergence check is performed (e.g. Newton’s method to solve an arbitrary polynomial equation as discussed above with respect to step 2A prong 1), and this is also considered as part of the mere instructions to apply a computer to implement the abstract idea, when this limitation is read in view of ¶ 2: “The simulation of physical systems often employs a mesh which is used in a discretization method ( e.g. finite volume method) to provide outputs for the simulation… The mesh representation may subsequently be used in setting up and solving equations for analysis using a discretization method (e.g. finite volume analysis) which may include numerical solution of partial differential equations based on boundary conditions. Other analysis methods can be used in other examples. Mesh-based analysis techniques are used widely in the fields of computational fluid dynamics (CFD), aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and other fields of engineering as well as manufacturing and fabrication processes, such as additive manufacturing processes” and ¶ 22 of the instant disclosure: “Solver module 158 can be any type of solver, such as a finite element solver, having a mesh-size independent stopping criterion. Solver module 158 can solve for a potential solution based on a set of equations, or physics 154, and a mesh model from mesh generator 156 generating solver results 160. Here the set of equations describe one or more characteristics being simulated”, wherein this is considered WURC in view of:
See ¶¶ 1-2 of the instant disclosure, as well as ¶¶ 18-20.
See Ansys (the instant assignee), article “What is Simulation”, accessed on July 26th, 2024, URL: developer(dot)ansys(dot)com/about-simulation#:~:text=The%20simulation%20software%20utilizes%20mathematical,dynamics%2C%20thermodynamics%2C%20and%20more(dot), ¶ 1: “Computer aided engineering (CAE) is a computer-based tool that enables users to simulate real-world scenarios and systems. The simulation software utilizes mathematical models and algorithms to simulate and analyze the behavior of a system over time. CAS can be used to study the behavior of mechanical systems, electrical systems, fluid dynamics, thermodynamics, and more. CAE is used in a variety of industries such as aerospace, automotive, electrical engineering, and architecture”.
See Comsol, “Finite Element Analysis (FEA) Software”, article in the “Multiphysics Cyclopedia”, Last modified: Feb. 21st, 2017, URL: www(dot)comsol(dot)com/multiphysics/fea-software – see the section “Inside FEA Software” including “The foundation of FEA software is formed by the laws of physics expressed in mathematical models… The mathematical models are discretized by the Finite Element Method (FEM), resulting in corresponding numerical models. The discretized equations are solved and the results are analyzed, hence the term finite element analysis…” and then see the section “Mathematical Model and Numerical Model” including “A mathematical model of a system can consist of one or several PDEs that describe the relevant laws together with boundary and initial conditions.”; also see the section “Results” for “An important task in postprocessing is to estimate the error in the numerical solution. As mentioned above, this can be achieved by solving the numerical model equations for different mesh sizes in order to estimate the convergence of the numerical solution.”
See previously cited COMSOL, “The Finite Element Method (FEM)”, 2019, “An Introduction to the Finite Element Method”, then see the section “Mesh Convergence” including “Mesh convergence is a simple method that compares approximate solutions obtained for different meshes. Ideally, a very fine mesh approximation solution can be taken as an approximation to the actual solution…The numerical model equations are solved for different mesh types and element sizes…The above plot shows that the relative error decreases with decreasing element size (h) for all elements. In this case, the convergence curve becomes steeper as the order of the basis functions (elements order)becomes higher. Note, however, that the number of unknowns in the numerical model increases with the element order for a given element…”
See previously cited COMSOL, “What Does Finite Element Analysis Software Bring?”, 2019, including ¶ 2 including: “…Once an FEA model is established and has been found useful in predicting real-life properties, it may generate the understanding and intuition to significantly improve a design and operation of a device or process…Most modern FEA software features methods for describing automatic control and incorporating such descriptions in mathematical and numerical models…” then see “Inside FEA Software” including: “The foundation of FEA software is formed by the laws of physics expressed in mathematical models. In the case of FEA, these laws consist of different conservation laws, laws of classical mechanics, and laws of electromagnetism…The mathematical models are discretized by the Finite Element Method…The discretized equations are solved and the results are analyzed, hence the term finite element analysis…The purpose of these descriptions is found in the study of the solution to the PDEs for a given system, which results in understanding a studied system and the ability to make predictions about…”, see the section “Mathematical Model and Numerical Model” as well, then see the section “The Processes Involved in Finite Element Analysis”, include seeing the subsection for the “Mesh” including: “Meshing is considered to be one of the most difficult tasks of preprocessing in traditional FEA. In modern FEA packages, an initial mesh may be automatically altered during the solution process in order to minimize or reduce the error in the numerical solution” then the subsection “Solution” including: “If creating the mesh is considered a difficult task, then selecting and setting the solvers and obtaining a solution to the equations (which constitute the numerical model) in a reasonable computational time is an even more difficult task…To accelerate the iterative solution process for linear systems, modern FEA packages use geometric or algebraic multigrid methods…For this purpose, a critical derived value in the model can be compared against a typical mesh element size. The computation is then repeated with different meshes (ideally, two other meshes that differ significantly from the current mesh and from each other). If things are in order with the numerical model, the order of accuracy can be estimated from the comparison. If the order of accuracy is positive, then the difference of the studied quantity between the two finest meshes can serve as an estimate of the truncation error for this quantity. It is not always possible to create several meshes that satisfy all requirements; a comparison between two radically different meshes will then have to serve as a substitute….”
See previously cited Bathe et al., “SOME PRACTICAL PROCEDURES FOR THE SOLUTION OF NONLINEAR FINITE ELEMENT EQUATIONS”, 1979, § 1: “During recent years the nonlinear finite element analysis of static and dynamic problems has been an area of growing interest in engineering. Various finite element computer programs are currently in use for the analyses of complex nonlinear problems [l]. Basically, these analyses involve three steps: the selection of a representative finite element model, the analysis of the model and the interpretation of the results. Surely, in engineering practice the selection of an appropriate finite element model and the corresponding interpretation of the results are crucial, but a reliable and accurate response prediction of the model is essential in order that the analysis results can be used with confidence.”
Previously cited Zigh et al., “Computational Fluid Dynamics Best Practice Guidelines for Dry Cask Applications”, 2012. Abstract: “Therefore, in cooperation with the Division of Spent Fuel Storage and Transportation of the Office of Nuclear Material Safety and Safeguards, the Office of Nuclear Regulatory Research developed this guide to provide practical advice for reviewing CFD methods used in vendor applications and for achieving high-quality CFD simulations of a dry cask.” And § 3.3: “Convergence is a major issue with the use of CFD software. Fluid mechanics is involved with nonlinear processes, dealing with inherently unstable phenomena, such as turbulence. CFD software simulates these physical processes and therefore is subject to the same issues as the processes it is trying to represent”
Previously cited Oberkampf et al., “Verification and Validation in Computation Fluid Dynamics”, 2002, page 33, ¶ 3: “Careful empirical assessment of iterative performance in a code is also important. In most practical application simulations, the equations are nonlinear and the vast majority of methods of solving these equations require iteration (for example, implicit methods for computing steady states in CFD). Iterations are typically required in two situations: (1) globally (over the entire domain) for boundary value problems, and (2) within each time step for initial-boundary value problems. Thus, iterative convergence is as much of an empirical issue as space-time grid convergence. The questions raised above for space-time convergence have exact analogs for iteration schemes: Does the iteration scheme converge and does it converge to the correct solution?””
Alauzet, Frédéric, et al. "3D transient fixed point mesh adaptation for time-dependent problems: Application to CFD simulations." Journal of Computational Physics 222.2 (2007): 592-623. Abstract, § 1 and § 2.2 including: “Here we consider a typical CFD simulation for flow problems modeled by the Euler equations” and § 3.3.1 including equation 16.
Misztal, I., D. Gianola, and L. R. Schaeffer. "Extrapolation and convergence criteria with Jacobi and Gauss-Seidel iteration in animal models." Journal of Dairy Science 71 (1988): 107-114. Abstract, then see the section “Convergence Criteria” including # 1-2.
Golovin, Sergey V., and Alexey N. Baykin. "Influence of pore pressure on the development of a hydraulic fracture in poroelastic medium." International Journal of Rock Mechanics and Mining Sciences 108 (2018): 198-208. Abstract, § 1, then see page 13, last paragraph.
Jaensson, N. O., M. A. Hulsen, and P. D. Anderson. "A comparison between the XFEM and a boundary-fitted mesh method for the simulation of rigid particles in Cahn–Hilliard fluids." Computers & Fluids 148 (2017): 121-136. Abstract, § 1, then see page 128, last paragraph
Tremblay, Pascal. 2-D, 3-D and 4-D anisotropic mesh adaptation for the time-continuous space-time finite element method with applications to the incompressible Navier-Stokes equations. Diss. University of Ottawa (Canada), 2008. § 1.1 including: “Over the past three decades, CFD has evolved to handle problems of increasing complexity. Numerical methods, including the finite difference (FDM), finite volume (FVM) and finite element (FEM) methods, were first developed to handle flows in fixed geometries…” and see §7.6, including equation 7.1 on page 154; also see § 7.4.
Saad, Yousef. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2003. See § 4.2 starting on page 114, include seeing § 4.2.1
Samuelsson, Klas. Numerical solution of pipe flow problems for generalized Newtonian fluids. No. LIU-TEK-LIC--1993-09. Linkoeping Univ.(Sweden). Dept. of Mathematics, 1993. See § 4.10, including page 38, ¶ 2, including the equations in ¶ 2.
Nomura et al., US 2014/0207428, ¶¶ 2-5, and ¶¶ 57, 61
Tsunoda et al., US 2012/0296616, ¶¶ 1-4 including: “…As well known, in a computerized fluid simulation, a flow domain or a three-dimensional space in which the fluid flows is split into subdomains (often called elements or cells) to generate a grid or mesh…”; also see ¶¶ 143-147
Welkie et al., US 2009/0083356, ¶¶ 3-5, also see ¶¶ 51, 60-61.
The claimed invention is directed towards an abstract idea of both a mathematical concept and a mental process without significantly more.
Regarding the dependent claims
Claim 24 is further limiting both the mental process and mathematical concept by excluding, in a negative limitation, the use of a “residual” calculation (¶ 18 of the instant disclosure)
Claims 31 and 38 are rejected under similar rationales.
Claim 25 is further limiting the mathematical calculation – see ¶ 26: “In this case, the Jacobian, J = aF is replaced with au J = J + w diagU), where w is a relaxation parameter applying a slope to the Jacobian. If the slope or the relaxation parameter is a large value, the iterations can generate potential solutions that overshoots which cause oscillations or a stall”, ¶ 30: “w is the relaxation factor,”, ¶ 35: “In one embodiment, the calculation is performed iteratively using a relaxation method.” – by reciting a mathematical method used to perform the mathematical calculation, wherein an example of the use of this method includes using a “relaxation parameter”/”relaxation factor” in the equations being calculated. To further clarify on “relaxation method” being a mathematical method, see Engeln-Müllges, G., Uhlig, F. (1996). Numerical Algorithms with C. Springer, Berlin, Heidelberg. See Chapter 5, including § 5.1 ¶ 2: “The procedures of this chapter such as the Jacobi and the GauB-Seidel method converge only linearly and very slowly in most practical cases on account of a growing Lipschitz constant L as the dimension n increases. For this reason, iterative methods are superior to the direct methods only in a very few special cases, namely if A is sparse, very large and structured in such a way that – if one were to employ one of the direct methods - the matrices would exceed the available memory. Their convergence can in general be accelerated by using a relaxation method based on either the Jacobi or GauB-Seidel method. Relaxation methods additionally require that at least some approximations to the eigenvalues of largest and smallest magnitude of the iteration matrix are known when using the GauB-Seidel method, or that the eigenvalue of largest magnitude only be known approximately when using the Jacobi method” – then see §§ 5.5-5.6.1 which give additional details, including on how a “relaxation coefficient” is used as part of this method (the Examiner notes the same variable is used for this parameter in the instant disclosure as cited above). Include seeing in § 5.5: “…In a relaxation method, one tries to improve the iterate x(v) by wz(v) instead of z(v) where w is called the relaxation coefficient.” Also see:
Previously cited Birken, “Numerical Methods for the Unsteady Compressible Navier-Stokes Equations”, 2012 § 5.5.1 including: “Finally, all these methods exist in a relaxation form, meaning that a relaxation parameter is added, which can lead to improved convergence speed.”; and § 8.5: “Various methods have been proposed to increase the convergence speed of the fixed point iteration by decreasing the interface error between subsequent steps, for example Relaxation [122, 118], Interface-GMRES [144] or ROM-coupling [213]. Relaxation means that after the fixed point iterate is computed, a relaxation step is added [see the equation for the step]…”
Previously cited Kelly, “Iterative Methods for Linear and Nonlinear Equations”, 1995, § 1.4 including ¶ 1
Previously cited Zigh et al., “Computational Fluid Dynamics Best Practice Guidelines for Dry Cask Applications”, 2012, § 3.3.2, ¶ 4: “The solution of the fully-coupled system of equations, and the inner loop of the noncoupled system, requires the solution of a set of linear, simultaneous equations; in other words, the inversion of a matrix. Except for small problems for which inversion by Gaussian elimination can be attempted, the solution algorithm is usually iterative. In fact, the success of finite-volume discretization schemes in CFD is largely because the algorithms produce diagonally dominant system matrices. Such matrices can be readily inverted using iterative methods. Many such methods have been derived, ranging from the classical Jacobi, Gauss-Seidel, successive-over-relaxation, and alternative direction implicit algorithms, through the more modern Krylov family of algorithms (e.g., conjugate-gradient or the more up-to-date multigrid and algebraic multigrid methods.) All such methods involve pivoting on the diagonal entry for each row of the matrix, and the success and speed of convergence of the iteration process essentially is governed by how much this term dominates over the sum total of the others in the row (supported by under-relaxation, if necessary) and the accuracy of the initial guess.”
Claims 32 and 39 are rejected under similar rationales.
Claim 26 is considered as a mental process step of a mental judgement/evaluation, such as aided by pen and paper, e.g. a person drawings, using pen and paper, a simple mesh of a simple physical system. For example, if the physical system is a square steel plate, a person may readily measure the dimensions of the steel plate with a ruler (a mental observation), and then use graph paper with pen/pencil to draw out a simple square mesh of the steel plate using two different sizes of mesh elements (wherein, to simplify the drawing, square mesh elements may be used, e.g. the first ones are the same size as the grid squares of the graph paper, and the second ones are ½ or 2x the size of the grid squares of the graph paper). This is also considered as a mathematical relationships, i.e. the mathematical geometrical relationships formed by dividing the geometry of a physical system into smaller parts of the geometry (e.g. dividing a square into a plurality of smaller squares, wherein the mesh is the mathematical relationships relating the dimensions of the smaller squares to the larger squares, akin to the example discussed above with respect to claim 22). To clarify on the BRI of this limitation, see fig. 6A-6C and ¶ 33: “Simulation 600 simulates a fluid flow in an x direction of a flat surface having a bump at approximately x=2.5 meters. Referring to Figures 6A-6C, the respective mesh sizes are 320 x 64, 640 x 128, and 1280 x 256”, wherein the Examiner further notes that 1) the claim does not recite what physical system is being simulated, and 2) a person would have been able to mentally generate a mesh for a moving truck, e.g. by taking an image of moving truck such as printed out on paper, and overlaying a piece of translucent graphing paper, wherein the squares of the graphing paper represent the mesh, wherein the person selects the graphing paper to be used according to the size of the grid cells on the graphing paper so the grid cells correspond to a desired mesh element size, wherein the person would then mentally observe the overlaid translucent graphing paper and thus readily be able to mentally visualize, in their own mind, the mesh. Should this be found to not be part of the abstract idea, then this would be a step of mere data gathering that is an insignificant extra-solution activity, wherein this is considered WURC in view of:
See previously cited COMSOL, “The Finite Element Method (FEM)”, 2019, “An Introduction to the Finite Element Method”, then see the section “Mesh Convergence” including “Mesh convergence is a simple method that compares approximate solutions obtained for different meshes. Ideally, a very fine mesh approximation solution can be taken as an approximation to the actual solution…The numerical model equations are solved for different mesh types and element sizes…The above plot shows that the relative error decreases with decreasing element size (h) for all elements. In this case, the convergence curve becomes steeper as the order of the basis functions (elements order)becomes higher. Note, however, that the number of unknowns in the numerical model increases with the element order for a given element…”
See previously cited COMSOL, “What Does Finite Element Analysis Software Bring?”, 2019, including ¶ 2 including: “…Once an FEA model is established and has been found useful in predicting real-life properties, it may generate the understanding and intuition to significantly improve a design and operation of a device or process…Most modern FEA software features methods for describing automatic control and incorporating such descriptions in mathematical and numerical models…” then see “Inside FEA Software” including: “The foundation of FEA software is formed by the laws of physics expressed in mathematical models. In the case of FEA, these laws consist of different conservation laws, laws of classical mechanics, and laws of electromagnetism…The mathematical models are discretized by the Finite Element Method…The discretized equations are solved and the results are analyzed, hence the term finite element analysis…The purpose of these descriptions is found in the study of the solution to the PDEs for a given system, which results in understanding a studied system and the ability to make predictions about…”, see the section “Mathematical Model and Numerical Model” as well, then see the section “The Processes Involved in Finite Element Analysis”, include seeing the subsection for the “Mesh” including: “Meshing is considered to be one of the most difficult tasks of preprocessing in traditional FEA. In modern FEA packages, an initial mesh may be automatically altered during the solution process in order to minimize or reduce the error in the numerical solution” then the subsection “Solution” including: “If creating the mesh is considered a difficult task, then selecting and setting the solvers and obtaining a solution to the equations (which constitute the numerical model) in a reasonable computational time is an even more difficult task…To accelerate the iterative solution process for linear systems, modern FEA packages use geometric or algebraic multigrid methods…For this purpose, a critical derived value in the model can be compared against a typical mesh element size. The computation is then repeated with different meshes (ideally, two other meshes that differ significantly from the current mesh and from each other). If things are in order with the numerical model, the order of accuracy can be estimated from the comparison. If the order of accuracy is positive, then the difference of the studied quantity between the two finest meshes can serve as an estimate of the truncation error for this quantity. It is not always possible to create several meshes that satisfy all requirements; a comparison between two radically different meshes will then have to serve as a substitute….” .
Previously cited Zigh et al., “Computational Fluid Dynamics Best Practice Guidelines for Dry Cask Applications”, 2012. Page 24, ¶ 3: “Although convergence of results as time-step or mesh size is reduced toward zero does not guarantee that the numerical solution is converging to the solution of the set of PDEs, it is a good indicator”
Previously cited Kelly, “Iterative Methods for Linear and Nonlinear Equations”, 1995. Page 31: “Experiment with different mesh sizes and preconditioners (Fast Poisson solver, Jacobi, and symmetric Gauss–Seidel).” Page 151, exercise 8.5.7: “Vary the convection coefficient in the convection-diffusion equation and the mesh size and report the results.”
Previously cited Koren, “COMPUTATIONAL FLUID DYNAMICS UNSTRUCTURED MESH OPTIMIZATION FOR THE SIEMENS 4TH GENERATION DLE BURNER”, 2015, abstract, then see § 3.2.1.1 including: “To track the oscillations of velocity and methane mass fraction during convergence process a number of monitor points were allocated to specific parts of the domain. The idea was to keep the steady state convergence process until the oscillations statistically stabilize. If the oscillations are less than the differences of a monitored quantity obtained from computing using different meshes then the steady state solution may be valid even for mesh convergence study.” - also see § 3.3 including: “To avoid that to some degree Ansys Fluent offers a powerful Grid Adaption tool which can refine and in some cases coarsen a mesh. That can be done manually or even automatically... This results in a drastic increase in grid cell count when splitting the whole domain but the advantage is that creating a new mesh is a matter of seconds while creating a new mesh in ICEM CFD can take one whole working day”
Claims 33 and 40 are rejected under similar rationales.
Claim 27 is generally linking to a variety of particular fields of use of “structural analysis, fluid analysis, thermal analysis, or aerodynamics analysis” (i.e. each one of these recitations is generally linking to a particular field of use), and further limiting the mental process and mathematical concept to a “steady state simulation solving…” (i.e. that the abstract idea is performed when the physical system is in steady-state/the conditions are not varying, e.g. for simulating a moving truck, this would be performing the calculations for the moving truck when the truck has a constant velocity, with no wind, with a single weight, etc.). Should this be found not to be abstract, then the Examiner submits that this would be generally linking to a particular field of use, akin to “iii. Limiting the use of the formula C = 2 (pi) r to determining the circumference of a wheel as opposed to other circular objects, because this limitation represents a mere token acquiescence to limiting the reach of the claim, Flook, 437 U.S. at 595, 198 USPQ at 199;… vi. Limiting the abstract idea of collecting information, analyzing it, and displaying certain results of the collection and analysis to data related to the electric power grid, because limiting application of the abstract idea to power-grid monitoring is simply an attempt to limit the use of the abstract idea to a particular technological environment, Electric Power Group, LLC v. Alstom S.A., 830 F.3d 1350, 1354, 119 USPQ2d 1739, 1742 (Fed. Cir. 2016);” as discussed in MPEP § 2106.05(h), as this is generally linking the abstract idea to be applied to “steady state simulation” instead of unsteady state simulation, e.g. transient simulation. These are also considered WURC in view of:
¶ 2 of the instant disclosure: “Mesh-based analysis techniques are used widely in the fields of computational fluid dynamics (CFD), aerodynamics, electromagnetic fields, civil engineering chemical engineering, naval architecture and other fields of engineering as well as manufacturing and fabrication processes, such as additive manufacturing processes”
COMSOL, “The Finite Element Method (FEM)”, 2019, section “The Finite Element Method from the Weak Formulation: Basis Functions and Test Functions” – “Assume that the temperature distribution in a heat sink is being studied, given by Eq. (8), but now at steady state, meaning that the time derivative of the temperature field is zero in Eq. (8).”, and page 14 ¶ 2: ‘The figure below depicts the temperature field around a heated cylinder subject to fluid flow at steady state. The stationary problem is solved twice: once with the basic mesh and once with a refined mesh controlled by the error estimation that is computed from the basic mesh.”
Previously cited Koren, “COMPUTATIONAL FLUID DYNAMICS UNSTRUCTURED MESH OPTIMIZATION FOR THE SIEMENS 4TH GENERATION DLE BURNER”, 2015, abstract, then see § 3.2.1: “It is often of engineering interest to obtain steady state RANS solution and so it is also in this study. Steady state solutions give a good picture of the reality needed in industry and are at the same time very cost effective therefore a lot of effort was put into attempt to acquire a steady state solution. However, as already mentioned it is not always possible for some cases to converge to the final steady state solution due to physical or numerical unsteady behaviour.” – also see § 3.2.1.1 including: “To track the oscillations of velocity and methane mass fraction during convergence process a number of monitor points were allocated to specific parts of the domain. The idea was to keep the steady state convergence process until the oscillations statistically stabilize. If the oscillations are less than the differences of a monitored quantity obtained from computing using different meshes then the steady state solution may be valid even for mesh convergence study.”
Previously cited Zigh et al., “Computational Fluid Dynamics Best Practice Guidelines for Dry Cask Applications”, 2012. § 3.3: “Convergence is a major issue with the use of CFD software. Fluid mechanics is involved with nonlinear processes, dealing with inherently unstable phenomena, such as turbulence. CFD software simulates these physical processes and therefore is subject to the same issues as the processes it is trying to represent. As such, it is not guaranteed that there will be a steady-state “converged” solution to a problem.”, also see page 25 ¶ 2, and § 3.3.5: “Iterative algorithms are used for steady-state solution methods and for procedures to obtain an accurate intermediate solution at a given time step in transient methods.”
Claims 34 and 41 are rejected under similar rationales.
Claim 28 is considered mere data gathering of a math concept of a set of equations into calculations, WURC in view of the evidence discussed above for the mere data gathering in the independent claims
Claims 35 and 42 are rejected under similar rationales.
The claimed invention is directed towards an abstract idea of both a mathematical concept and a mental process without significantly more.
Conclusion
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/David A Hopkins/Primary Examiner, Art Unit 2188