Notice of Pre-AIA or AIA Status
Claims 1, 4-9, 12-17 and 20-24 are currently presented for Examination.
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .
Continued Examination Under 37 CFR 1.114
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and the fee set forth in 37 CFR 1.17(e) has been timely paid, the finality of the previous Office action has been withdrawn pursuant to 37 CFR 1.114.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 09/09/2026 has been entered.
Response to Amendment
The amendment filed on 09/09/2026 has been entered and considered by the examiner. By the amendment, claims 1, 4, 9, 12, 17 and 20 are amended and claims 3, 11 and 19 are cancelled. The 101 rejection is still maintained. New 112(b) rejection is added. See office action for detail.
Applicant 101 arguments
The Amended Claims are Not Properly Characterized as Merely a Mental Process. The Independent Claims Integrate Any Recited Mathematical Concepts Into a Practical Application and Recite an Improvement to Computer Modeling and FEA Technology (page 16-17)
Examiner response
Examiner respectfully disagrees. The rejection did not depend on the metal-process grouping only. The claim independently recites mathematical concepts including barycentric coordinate relationships, Bernstein bases, continuity constraints, nullspace calculations, coefficient restrictions, index-support relationships, and normalization. Whether those operations can practically be performed mentally does not determine whether they fall within the mathematical concepts grouping. 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). Accordingly, Applicant's arguments regarding the mental process grouping do not overcome the mathematical concepts finding under Step 2A, Prong One.
While the applicant argues that math is integrated into a practical computer modeling application, an examiner respectfully disagrees. The claim merely says "employ [the math] ... to perform finite element analysis of the CAD model." It does not specify how the computer hardware is altered, nor does it provide a specific engineering solution. It simply states the mathematical desired result (local modification of cell size, polynomial degree, and intercell continuity) and uses a generic computer to calculate it. 1. The specification admits that FEA, CAD models, and existing spline technologies (like NURBS and T-splines) are already "prevailing industrial spline representations." Merely substituting a new mathematical framework (U-splines) into standard, well-understood, and conventional computer software routines (FEA and CAD manipulation) does not provide "significantly more" than the abstract mathematical concept itself.
Applicant arguments
The specification identifies a specific technological problem. (page 18)
Examiner response
Applicant relies on paragraph [0006]-[0008] and [0018]-[0020] which describe the difficulty and numerical instability of constructing a sparse positive spline basis through a global nullspace solution. Examiner acknowledges that the specification describes a substantial computational problem. Identifying such a problem, however, does not by itself establish eligibility. The inquiry remains whether the claim reflects a technological implementation of the solution or an improved mathematical procedure executed by a computer. Here, the identified problem concerns determining the basis for a mathematical constrain system. The claimed solution principally reorganizes that determination through cell constraints systems, recursive basis-vector construction, graph-based support selection and support-specific nullspace calculations. Those requirements specify the mathematical procedure more particularly, but particularity alone does not establish integration into a technological application.
Applicant arguments
The amended claim recites the particular computational mechanism that provides the improvement.(page 18-19)
Examiner response
Applicant argues that recursive partitioning, local nullspace construction, core graphs, and rank-one nullspace solving provide the disclosed computational improvement. Examiner respectfully disagrees. Recursive portioning specifies the organization of the continuity equations; the coefficient and adjacency restrictions specify permissible basis vectors; the core graph determines mathematical index support; and the nullspace and normalization operations determine the resulting basis functions. Their coordinated operation constructs spline representation. The claim’s additional implementation requirements nevertheless remain processor execution, storage, and employment of the resulting in CAD modeling and FEA. The claim does not specify how the constructed basis changes the operation of a CAD modeling process or an FEA solver beyond supplying the mathematical representation used by those processes. Furthermore, Applicant’s discussion relies in part on features not expressly required by claim 1 including integer-only processing before support determination, particular admissibility conditions, elimination of knot vectors and T-meshes and expressly bounded local problem sizes. Those disclosures inform interpretation but cannot automatically be treated as additional claim requirements. The office action does not require numerical speed improvement or new physical hardware. Rather, it finds that the presently claimed implementation does not establish a technological improvement distinguishable from the improved mathematical construction itself.
Applicant arguments
Ex parte Desjardins supports eligibility of the amended claim. (page 20)
Examiner response
Examiner respectfully disagrees. Unlike Ex parte Desjardins—where the claims directly modified machine learning parameters to solve a specific computing problem (catastrophic forgetting in neural networks), the instant claim of constructing U-spline meshes for CAD/CAE applications represents a mathematical modeling technique applied to an end-use field. The claimed steps—recursive partitioning, local-nullspace construction, and rank-one nullspace computation—describe purely mathematical manipulations of data structures (constraint matrices and nullspaces) rather than an improvement to computer execution capability or data storage mechanics. The instant claim recites using the math ("employ the refined U-spline mesh... to smoothly and accurately represent...") but does not recite a specific, physical, or hardware-based improvement to computer functionality itself. The specification [0117] confirms that the "U-spline technology" is a mathematical framework designed to satisfy technical objectives (h-refinement, p-refinement) that were previously done by other mathematical formats (NURBS, T-splines). It improves the math, not the physical computer hardware. The claim applies the mathematical concept to the field of "CAD model" analysis. However, judicial precedent (like Parker v. Flook) dictates that limiting an abstract idea to a specific technological field (here, CAD/FEA engineering software) does not provide a practical application. The claim does not improve the functioning of the computer itself; rather, it uses the computer as a generic tool to execute the U-spline math.
Applicant arguments
Enfish likewise supports eligibility because the improvement resides in a particular logical/computational architecture. (see page 21)
Examiner response
Applicant argues that Enfish establishes that software implementation may reside in local structures rather than new hardware. Examiner agrees with that principle. Under the Alice Step 1 framework, decomposing a global continuity problem into localized cell systems, building localized nullspaces, and solving rank-one nullspace problems are mathematical operations. The claim does not improve the computer's hardware, operating system, or fundamental processing capacity. Instead, it describes a more efficient mathematical method for solving a geometric or physical simulation problem (U-splines). Mathematical formulas and algorithms do not become patent-eligible simply because they are broken down into smaller, more manageable mathematical steps (e.g., local vs. global). The computational organization described by Applicant is a mathematical methodology. The problems of 'computational expense' and 'numerical instability' identified in the Specification are inherent limitations of mathematical algorithms, not computer hardware. Because the generic computer is merely used as a tool to perform these localized mathematical steps, the claim does not recite a patent-eligible improvement to computer functionality." Thus, claim 1 does not establish that its organization improves computer functionality or CAD/FEA operation beyond calculating and supplying the resulting spline representation.
Applicant arguments
McRO further confirms that the use of specific mathematical rules to improve a technological process is not disqualifying. (page 21-22)
Examiner response
Examiner respectfully disagrees. The claims recite generic mathematical calculations for constructing splines rather than a specific, non-abstract technological improvement that changes how a computer functions. Unlike the unconventional, specific rules in McRO, Inc. v. Bandai Namco Games America Inc. that replaced subjective animator judgments with concrete phoneme-to-timing rules to automate a specific 3D animation sub-process, the instant claims merely set forth generalized mathematical concepts—namely, localized constraint and nullspace architectures for spline calculations—without tying those mathematical operations to a concrete technological output or an unconventional operational improvement in computer capabilities. Simply labeling the algorithm a "fundamentally different method" in the specification does not make the math patent eligible. Claim 1 detailed rules govern coefficient, continuity relationships, recursive nullspace construction and basis support. The CAD model representation and FEA limitations do not further specify how those rules are implemented within an improved modeling or analysis process. The rejection therefore does not disregard the claimed rules. It finds their claimed application insufficient to establish the technological improvement asserted by Applicant.
Applicant arguments
The amended CAD/FEA limitations are not merely a nominal field-of-use limitation. (see page 22-23)
Examiner response
Applicants argue that requiring CAD representation and FEA requires beyond merely identifying a field of use. The instant claim recites using the math ("employ the refined U-spline mesh... to smoothly and accurately represent...") but does not recite a specific, physical, or hardware-based improvement to computer functionality itself. The specification [0117] confirms that the "U-spline technology" is a mathematical framework designed to satisfy technical objectives (h-refinement, p-refinement) that were previously done by other mathematical formats (NURBS, T-splines). It improves the math, not the physical computer hardware. The claim applies the mathematical concept to the field of "CAD model" analysis. However, judicial precedent (like Parker v. Flook) dictates that limiting an abstract idea to a specific technological field (here, CAD/FEA engineering software) does not provide a practical application. The claim does not improve the functioning of the computer itself; rather, it uses the computer as a generic tool to execute the U-spline math. The additional elements of employing the refined U-spline mesh and the U-spline basis to smoothly and accurately represent a CAD model amount to nothing more than a field of use limitation. (MPEP 2106.05(h)) The limitation merely restricts the use of the data (U-spline mesh) to a particular technological environment (CAD/CAE). This is a "token acquiescence to limiting the reach of the claim," as described in Flook, rather than an improvement to the CAD process itself. It describes the "field of use" (CAE analysis) rather than a "technical solution to a technical problem".
Applicant arguments
Applicant argues that under Step 2A, Prong Two, the conventionality of the processors or storage media cannot defeat eligibility, and that the claims reflect an improvement to CAD/FEA technology by applying localized computational steps for U-spline construction.
Examiner response
The Examiner agrees that the question of whether a component is "well-understood, routine, or conventional" is generally reserved for Step 2B. However, Step 2A, Prong Two still requires the claim as a whole to integrate the exception into a practical application by delivering a technological solution to a technological problem. It is the Examiner position that the instant claims are directed to mathematical improvement rather than a technological improvement. The "localized computational steps" yield a more efficient mathematical representation (the U-spline). Furthermore, the recitation of "CAD" or "FEA" merely defines the field of use for this mathematical optimization. Limiting an abstract idea to a particular technological environment or field of use does not constitute a practical application under Step Prong Two. The computer simply executes the mathematical steps to generate data, and the data is used in CAD/FEA exactly as any other spline data would be used. The underlying functioning of the CAD/FEA software environment or the computer hardware itself is not altered. Because the claim amounts to nothing more than calculating an improved mathematical structure within its native technological environment, it remains directed to an abstract idea, and the rejection is maintained.
Applicant arguments
Applicant argues that the ordered combination supplies significantly more because it differs from a global nullspace approach in step 2B. (see page 23-25)
Examiner response
Examiner respectfully disagrees. Examiner considered the complete sequence in claim 1: constrain portioning, constrained basis-vector construction, core-graph support determination, support-specific nullspace solving, normalization, storage, CAD representation and FEA are mathematical concepts. The independent claims explicitly recite mathematical operations, including "calculating a nullspace," "manipulating Bernstein coefficients," and solving "rank-one nullspace problems." These elements constitute mathematical formulas, equations, and algorithms. The core of the claimed invention is an algorithmic optimization of a geometric and algebraic problem (U-spline generation). Modifying the mathematical approach from a "global constraint-matrix technique" to "incremental local operations" changes the specific mathematical steps being performed, but it does not change the fundamental nature of the claim: it remains a mathematical concept. Further, BASCOM cited by applicant requires consideration of whether the additional implementation elements, in combination with the exception, provide a nonconventional technological arrangement. Here, the processor executes the construction, storage retains the result, and the downstream CAD/FEA requirements broadly employ that result. Also, the applicant arguments regarding withdraw the prior art rejection- see MPEP 2106.05 that states Specifically, lack of novelty under 35 U.S.C. 102 or obviousness under 35 U.S.C. 103 of a claimed invention does not necessarily indicate that additional elements are well-understood, routine, conventional elements. Because they are separate and distinct requirements from eligibility, patentability of the claimed invention under 35 U.S.C. 102 and 103 with respect to the prior art is neither required for, nor a guarantee of, patent eligibility under 35 U.S.C. 101.
Furthermore, under Step 2B, the claim fails to recite significantly more than the abstract mathematical concept itself. The recitation of generic computer hardware or a generalized 'computational arrangement' to execute the mathematical steps amounts to routine, conventional computer implementation. Accordingly, the rejection under 35 U.S.C. § 101 as being directed to an abstract idea is maintained."
Claim Rejections - 35 USC § 112
The following is a quotation of 35 U.S.C. 112(b):
(b) CONCLUSION.—The specification shall conclude with one or more claims particularly pointing out and distinctly claiming the subject matter which the inventor or a joint inventor regards as the invention.
The following is a quotation of 35 U.S.C. 112 (pre-AIA ), second paragraph:
The specification shall conclude with one or more claims particularly pointing out and distinctly claiming the subject matter which the applicant regards as his invention.
Claims 1, 4-9, 12-17 and 20-24 are rejected under 35 U.S.C. 112(b) or 35 U.S.C. 112 (pre-AIA ), second paragraph, as being indefinite for failing to particularly point out and distinctly claim the subject matter which the inventor or a joint inventor (or for applications subject to pre-AIA 35 U.S.C. 112, the applicant), regards as the invention.
The term “accurately” in claim 1, 9 and 17 is a relative term which renders the claim indefinite. The term “accurately” is not defined by the claim, the specification does not provide a standard for ascertaining the requisite degree, and one of ordinary skill in the art would not be reasonably apprised of the scope of the invention. The term “accurately” in smoothly and accurately represent a CAD model that lacks an identified objective standard in the supplied disclosure. It is therefore unclear what degree of deviation between the U-spline representation, and the CAD model satisfied the limitation.
Claim 4-8, 12-16 and 20-24 are dependent claims do not overcome the deficiency of independent claims and thus rejected as well.
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.
5. Claims 1, 4-9, 12-17 and 20-24 are rejected under 35 U.S.C. 101 because the claim are directed to an abstract idea without significantly more.
(Step 1) Is the claims to a process, machine, manufacture, or composition of matter?
Claim 1 and 4-8 recites “system… comprising non-transitory computer readable storage media”., which falls into the one of the statutory categories, i.e., manufacture or machine.
Claims: 9 and 12-16 are directed method or process, which falls on the one of the statutory category.
Claim 17 and 20-24 recites “a computer program product… comprising non-transitory computer readable data storage”, which falls into the one of the statutory categories, i.e., manufacture or machine.
(Step 2A) (Prong 1) Is the claim directed to a law of nature, a natural phenomenon, or an abstract idea? (Judicially recognized exceptions)?
The following limitations under broadest reasonable interpretation recite abstract ideas:
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).
Claim 1, 9 and 17 recites:
construct a mesh which comprises a plurality of cells; (Under the broadest reasonable interpretation, this limitation covers mental process including an evaluation or judgment that could be performed in the human mind or with the aid of pencil and paper therefore falls within the “Mental Process” grouping of abstract ideas. A mesh itself is an abstraction, a mathematical representation of a complex shape or domain, divided into simpler components called cells.)
assign a coordinate system to each cell in the mesh, wherein each coordinate system is barycentric or formed from tensor products of barycentric coordinate systems; (Under the broadest reasonable interpretation, this limitation covers mental process including an evaluation or judgment that could be performed in the human mind or with the aid of pencil and paper therefore falls within the “Mental Process” grouping of abstract ideas. Also, these limitations are abstract mathematical constructs that define a point's location within a simplex (like a triangle or tetrahedron) by referencing the vertices of that simplex. So, it falls under the mental process or mathematical concepts of abstract ideas)
assign a parametric length to each edge of each cell in the mesh, wherein the parametric lengths satisfy the conditions for seamless similarity maps; (While the process of defining these lengths and applying them to a mesh certainly involves abstract ideas about shape and geometry, the core process is one of applying mathematical rules and calculations. The process involves computations based on mathematical algorithms and concepts, such as those related to conformal mapping. So, it falls under the mathematical concepts of abstract ideas)
assign a basis to each cell in the mesh, wherein each basis of each cell is a Bernstein or Bernstein-like basis; (The Bernstein basis itself is a well-defined mathematical concept, with specific properties, formulas, and applications, particularly in approximating functions and representing curves and surfaces.
So, it falls under the mathematical concepts of abstract ideas)
assign a minimum desired continuity to each interface between adjacent cells in the mesh;( In the context of meshes in specification, it refers to how smoothly the solution of a numerical simulation transitions across element boundaries. So, it falls under the mathematical concepts of abstract ideas)
construct a system of continuity constraints, termed the global system of constraints, from each interface in the mesh, wherein the global system of constraints is derived from a coordinate system, parametric lengths, and basis for each cell and the continuity associated with each interface; While the process of setting up and visualizing these constraints might involve a mental process of abstract ideas, the core principles and techniques used to construct the global system are based on rigorous mathematical definitions and procedures. So, it falls under the mathematical concepts of abstract ideas)
construct a refined U-spline mesh by partitioning the global system of constraints associated with the U-spline mesh into cell systems of constraints wherein: a cell system of constraints associated with cells of maximum dimension is empty; and a cell system of constraints for each cell of lower dimension is formed recursively from the cell systems of constraints of adjacent cells of higher dimension until an interface dimension is reached. (The construction of the mesh relies on understanding and manipulating abstract concepts related to partitioning constraints and recursive formation of systems. This involves mental processing of these abstract ideas to implement the mesh refinement process. U-splines are a mathematical approach to constructing spline bases used in CAD and CAE for representing smooth objects. The method described involves mathematically defined concepts like "global system of constraints," "cell systems of constraints," and "recursion" to refine the mesh based on continuity requirements. So, it falls under the combination of mental process and mathematical concepts of abstract ideas.)
construct a U-spline basis for the refined U-spline mesh by: constructing a set of vertex basis vectors for each vertex in the refined U-spline mesh by computing basis vectors for a nullspace of the cell system of constraints associated with each cell in the refined U-spline mesh, wherein: a basis vector for the nullspace has nonnegative coefficients when expressed in Bernstein form; a basis vector for the nullspace has a minimal number of nonzero coefficients in the Bernstein form; every nonzero coefficient in a basis vector for the nullspace is within one index unit of at least one other nonzero coefficient when other nonzero coefficients are required; a basis vector for the nullspace is computed from associated interfaces first; and a basis vector for the nullspace for lower-dimensional cells is computed by analyzing a set of basis vectors associated with the nullspaces of adjacent cells of dimension one greater; determining an index support of each U-spline basis vector by constructing a corresponding core graph; determining a basis vector of each U-spline basis function by solving a rank one null space problem for the corresponding index support; and normalizing the set of U-spline basis vectors to determine a set of U-spline basis functions. (The steps rely on linear algebra (computing nullspace, solving a rank-one null space problem) and polynomial representations (Bernstein form coefficients). So, it falls under the mathematical concepts of abstract ideas.)
employ the refined U-spline mesh and the U-spline basis to perform finite element analysis of the CAD model, wherein the refined U-spline mesh permits local modification of cell size, polynomial degree, and intercell continuity; (A "U-spline mesh" and a "U-spline basis" are mathematical constructs (specifically, piecewise polynomial functions and geometric topology representations used to discretize a space). Claim further recites “wherein the refined U-spline mesh permits local modification of cell size, polynomial degree, and intercell continuity" describes the mathematical properties and parameters inherent to the U-spline equation itself, rather than a physical or concrete step. A spline basis and mesh are mathematical algorithms used to approximate geometric space, and FEA is a series of matrix algebra calculations. Because the claim focuses on manipulating these mathematical relationships to simulate a CAD model, it is directed to a mathematical concept.)
Step 2A, Prong 2: Does the claim recite additional elements that integrate the judicial exception
In accordance with Step 2A, Prong 2, the judicial exception is not integrated into a practical application. In particular, claim 1, 9 and 17 recites the additional elements of store the refined U-spline mesh and the U-spline basis in durable data storage is recited at a high level of generality (i.e., as a general means of storing data), and falls under the insignificant extra solution activity. (See MPEP 2106.05(g)) The additional elements of employ the refined U-spline mesh and the U-spline basis to smoothly and accurately represent a CAD model in claim 1, 9 and 17 amount to nothing more than a field of use limitation. (MPEP 2106.05(h)) The limitation merely restricts the use of the data (U-spline mesh) to a particular technological environment (CAD/CAE). This is a "token acquiescence to limiting the reach of the claim," as described in Flook, rather than an improvement to the CAD process itself. It describes the "field of use" (CAE analysis) rather than a "technical solution to a technical problem". The additional elements of one or more computer processors; and non-transitory computer readable storage media having stored therein computer-executable instructions which, when executed by the one or more computer processors in claim 1 and a computer program product for constructing a U-spline mesh in computer aided design (CAD) or computer aided engineering (CAE), the system comprising: non-transitory computer readable data storage having stored therein computer-executable instructions which, when executed by one or more computer processors within a computing system in claim 17 are amounts to no more than mere instructions to apply the exception using generic computer components. (MPEP 2106.05(f); Thus, a system or method for constructing a U-spline mesh in computer aided design (CAD) or computer aided engineering (CAE) as discussed on MPEP 2106.05(h). Therefore, claims 1, 9 and 17 are directed to an abstract idea.
Step 2B: Does the claim recite additional elements that amount to significantly more than the judicial exception?
The claims do not include additional elements that are sufficient to amount to significantly more than the judicial exception. As discussed above with respect to the integration of the abstract idea into a practical application, the additional elements of store the refined U-spline mesh and the U-spline basis in durable data storage is recited at a high level of generality (i.e., as a general means of storing data), and falls under the insignificant extra solution activity and is well-understood, routine or conventional. ((See MPEP 2106.05(d)(II) (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; The additional elements of employ the refined U-spline mesh and the U-spline basis to smoothly and accurately represent a CAD model in claim 1, 9 and 17 amount to nothing more than a field of use limitation. (MPEP 2106.05(h)) The limitation merely restricts the use of the data (U-spline mesh) to a particular technological environment (CAD/CAE). This is a "token acquiescence to limiting the reach of the claim," as described in Flook, rather than an improvement to the CAD process itself. It describes the "field of use" (CAE analysis) rather than a "technical solution to a technical problem". The additional elements of one or more computer processors; and non-transitory computer readable storage media having stored therein computer-executable instructions which, when executed by the one or more computer processors in claim 1 and a computer program product for constructing a U-spline mesh in computer aided design (CAD) or computer aided engineering (CAE), the system comprising: non-transitory computer readable data storage having stored therein computer-executable instructions which, when executed by one or more computer processors within a computing system in claim 17 are amounts to no more than mere instructions to apply the exception using generic computer components. The statement that the method is performed by computer does not satisfy the test of “inventive concept.” See Alice Corp. Pty. Ltd. v. CLS Bank Int’l, 573 U.S. 208, 134 S. Ct. 2347, 2360 (2014). (MPEP 2106.05(f); Claim therefore, when taken as a whole, still does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception. Thus, a method for controlling a coordinate measuring machine is no more than generally linking to the field of use as discussed on MPEP 2106.05(h). Therefore, claims 1, 9 and 17 are directed to abstract idea.
Claim 4, 12 and 20 further recites wherein a third refined U-spline mesh is constructed by constructing a special set of basis vectors, termed the boundary set, for each set of vertex basis vectors in the mesh by:
determining a set of basis vectors associated with cell systems of constraints adjacent to the vertex, wherein nonzero coefficient indices in the basis vectors appear in at least one basis vector in the set of vertex basis vectors; determining appropriate charts for each cell adjacent to a vertex; forming equivalence classes of cell basis vectors adjacent to a vertex, wherein the equivalence class is determined through an appropriate projection; finding a basis vector from each equivalence class, wherein the nearest projected index point of the basis vector is furthest from the vertex in all appropriate charts; and forming a set containing the most distant basis vectors from each equivalence class. This process relies heavily on mathematical concepts such as basis vectors, projections, and geometric relationships between cells and vertices in a mesh. It involves understanding and applying principles from linear algebra (basis vectors) and geometry (charts, projections). So, it falls under the mathematical concepts of abstract ideas. The claim does not include any additional element; thus, it does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception.
Claim 5, 13 and 21 further recites wherein a fourth refined U-spline mesh is constructed by constructing nonzero coefficients of a single basis vector for a nullspace of the global system of constraints from the vertex basis vectors by: Step 1: forming a set, termed the basis index set, wherein each index in the basis index set is contained in at least one vertex basis vector and the associated Bernstein coefficient is nonzero;
Step 2: given one vertex basis vector, using the boundary set to determine the vertex basis vectors of all adjacent vertices whose boundary sets are aligned and adding these to the basis index set; Step 3: repeating Steps 1-2 until no adjacent vertices contain vertex basis vectors with aligned boundaries that are not in the basis index set; Step 4: taking the basis index set produced in Steps 1-3 and computing values for all Bernstein coefficients associated with the indices in the basis index set, wherein the values are determined from the global system of constraints while also enforcing positivity of all nonzero Bernstein coefficients. The process is a blend of applying mathematical concepts (nullspaces, Bernstein coefficients, constraints) within a structured, computational algorithm for generating the U-spline mesh. So, it falls under the mathematical concepts of abstract ideas. The claim does not include any additional element; thus, it does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception.
Claim 6, 14 and 22 further recites wherein Steps 14 are applied repeatedly until all unique basis vectors for the nullspace of the global system of constraints have been generated to form a U-spline basis. The process is a blend of applying mathematical concepts (nullspaces, Bernstein coefficients, constraints) within a structured, computational algorithm for generating the U-spline mesh. So, it falls under the mathematical concepts of abstract ideas. The claim does not include any additional element; thus, it does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception.
Claim 7, 15 and 23 further recites wherein individual basis vectors are scaled so that the sum over all basis vectors in Bernstein form produces a vector with every entry equal to 1. This is a mathematical concept where a set of functions (like Bernstein basis polynomials) sum to 1 over a given domain. So, it falls under the mathematical concepts of abstract ideas. The claim does not include any additional element; thus, it does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception. The claim does not include any additional element; thus, it does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception.
Claim 8, 16 and 24 further recites wherein the U-spline mesh, basis of each cell, and minimum desired continuity of each interface between adjacent cells are modified such that v-separation, p-separation, and v-grading are satisfied. It involves the modifications described in the query involve applying and manipulating mathematical concepts related to splines and meshes to achieve desired outcomes or properties. So, it falls under the mathematical concepts of abstract ideas. The claim does not include any additional element; thus, it does not integrate the judicial exception into a practical application nor amount to significantly more than the judicial exception.
Allowable Subject Matter
7. Claim 1, 4-9, 12-17 and 20-24 is objected to, but would be allowable if rewritten to overcome the 101 and 112 rejections of the claims. The following is a statement of reasons for the indication of allowable subject matter:
All the cited references in previous office action taken either together or in combination with the prior art of record fail to disclose instructions, including:
Claim limitation
construct a mesh which comprises a plurality of cells;
assign a coordinate system to each cell in the mesh, wherein each coordinate system is barycentric or formed from tensor products of barycentric coordinate systems;
assign a parametric length to each edge of each cell in the mesh, wherein the parametric lengths satisfy the conditions for seamless similarity maps; assign a basis to each cell in the mesh, wherein each basis of each cell is a Bernstein or Bernstein-like basis;
assign a minimum desired continuity to each interface between adjacent cells in the mesh;
construct a system of continuity constraints, termed the global system of constraints, from each interface in the mesh, wherein the global system of constraints is derived from a coordinate system, parametric lengths, and basis for each cell and the continuity associated with each interface;
construct a refined U-spline mesh by partitioning the global system of constraints associated with the U-spline mesh into cell systems of constraints wherein:
a cell system of constraints associated with cells of maximum dimension is empty; and
a cell system of constraints for each cell of lower dimension is formed recursively from the cell systems of constraints of adjacent cells of higher dimension until an interface dimension is reached;
construct a U-spline basis for the refined U-spline mesh by:
constructing a set of vertex basis vectors for each vertex in the refined U-spline mesh by computing basis vectors for a nullspace of the cell system of constraints associated with each cell in the refined U-spline mesh, wherein: a basis vector for the nullspace has nonnegative coefficients when expressed in Bernstein form;
a basis vector for the nullspace has a minimal number of nonzero coefficients in the Bernstein form;
every nonzero coefficient in a basis vector for the nullspace is within one index unit of at least one other nonzero coefficient when other nonzero coefficients are required;
a basis vector for the nullspace is computed from associated interfaces first; and a basis vector for the nullspace for lower-dimensional cells is computed by analyzing a set of basis vectors associated with the nullspaces of adjacent cells of dimension one greater;
determining an index support of each U-spline basis vector by constructing a corresponding core graph;
determining a basis vector of each U-spline basis function by solving a rank one null space problem for the corresponding index support; and
normalizing the set of U-spline basis vectors to determine a set of U-spline basis functions;
store the refined U-spline mesh and the U-spline basis in durable data storage; and
employ the refined U-spline mesh and the U-spline basis to smoothly and accurately represent a CAD model and to perform finite element analysis of the CAD model, wherein the refined U-spline mesh permits local modification of cell size, polynomial degree, and intercell continuity.
Conclusion
6. Claims 1, 4-9, 12-17 and 20-24 are rejected.
The prior art made of record and not relied upon is considered pertinent to applicant's disclosure:
US 20190303531 A1 Urick et al.
Discussing a mechanism for reconstructing trimmed surfaces whose underlying spline surfaces intersect in model space, so that the reconstructed version of each original trimmed surface is geometrically close to the original trimmed surface, and so that the boundary of each respective reconstructed version includes a model space trim curve that approximates the geometric intersection of the underlying spline surfaces.
Toshniwal, Deepesh, Hendrik Speleers, and Thomas JR Hughes. "Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations." Computer Methods in Applied Mechanics and Engineering 327 (2017): 411-458.
Discussing a novel framework for geometric modeling and isogeometric analysis on unstructured quadrilateral meshes.
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/PURSOTTAM GIRI/Examiner, Art Unit 2186
