1. A subparametric finite element analysis system (100) comprising: a processor (102); a memory (104) operatively coupled to the processor (102), wherein the processor (102) is configured to execute instructions stored in the memory to: generate a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface; retrieve from the generated high-order finite element mesh, node coordinates, element connectivity, and geometric information required for subparametric finite element computations; modify coordinates of boundary nodes of the generated high-order finite element mesh to enable subparametric transformations; generate shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh; compute element-level governing equations using the generated shape functions; compute, for each finite element, a Jacobian matrix and a determinant of the Jacobian matrix based on the computed element-level governing equations, the computed Jacobian matrix being used to validate the geometric transformations; compute element stiffness matrices using subparametric finite element formulations based on the computed Jacobian matrix; assemble a global stiffness matrix from the element stiffness matrices using parallel computation executed on a CUDA-enabled graphics processing unit (130) operatively coupled to the processor (102); and apply physical boundary conditions to the global stiffness matrix using parallel computation executed on the CUDA-enabled graphics processing unit (130) to produce a numerical solution of a finite element problem defined on the computational domain.

2. The system (100) as claimed in claim 1, wherein the Jacobian matrix and determinant of the Jacobian matrix are computed for each element using subparametric transformations to map reference element coordinates to physical element coordinates.

3. The system (100) as claimed in claim 1, wherein the processor (102) is further configured to perform symbolic computation to compute the element-level governing equations, wherein to perform the symbolic computation, the processor is configured to: derive element-level equations using Lagrange shape functions generated for each finite element; compute the Jacobian matrix and determinant of the Jacobian for each element based on the derived element-level equations; validate the geometric transformations of the high-order finite element mesh using the computed Jacobian and determinant; and generate symbolic representations of the element stiffness matrices prior to numerical evaluation and assembly into the global stiffness matrix.

4. The system (100) as claimed in claim 1, wherein the processor (102) is further configured to perform adaptive refinement of the high-order finite element mesh based on geometric features or element-specific parameters received from the node coordinates, element connectivity, and geometric information extracted from the generated high-order finite element mesh.

5. The system (100) as claimed in claim 1, wherein element stiffness matrices are computed in parallel across multiple finite elements on the CUDA-enabled graphics processing unit (130).

6. The system (100) as claimed in claim 1, wherein the processor (102) utilises the retrieved geometric information, including node coordinates and element connectivity, to generate the boundary node modifications required for subparametric finite element computations.

7. The system (100) as claimed in claim 1, wherein the processor (102) is configured to define the shape functions corresponding to the selected polynomial order of the modified high-order finite element mesh, wherein the processor (102): selects a polynomial order for the finite element; defines the shape functions for each node of the modified high-order finite element; evaluates the shape functions at each integration point of the finite element; and stores the evaluated shape functions in structured arrays or data matrices for element-level computation.

8. A method (300) for performing subparametric finite element analysis, comprising the steps of: generating (302), by a processor, a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface; retrieving (304), by the processor, from the generated high-order finite element mesh node coordinates, element connectivity, and geometric information required for subparametric finite element computations; modifying (306), by the processor, coordinates of boundary nodes of the generated high-order finite element mesh to enable geometric transformations; defining (308), by the processor, shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh, the step comprising: selecting a polynomial order for each finite element, defining Lagrange shape functions for each node of the modified high-order finite element, evaluating the shape functions at each integration point of the finite element, and storing the evaluated shape functions in structured arrays or data matrices; computing (310), by the processor, element-level governing equations using the defined shape functions; computing (312), by the processor, for each finite element, a Jacobian matrix and a determinant of the Jacobian matrix based on the computed element- equations using Symengine’s API, the Jacobian matrix being used to validate the geometric transformations; computing (314), by the processor, element stiffness matrices using subparametric finite element formulations based on the computed Jacobian matrix; assembling (316), by the processor, a global stiffness matrix from the element stiffness matrices using parallel computation executed on a CUDA-enabled graphics processing unit; and applying (318), by the processor, physical boundary conditions to the global stiffness matrix using parallel computation executed on the CUDA-enabled graphics processing unit to produce a numerical solution of the finite element problem defined on the computational domain.

9. The method as claimed in claim 8, wherein computing the element-level governing equations comprises performing symbolic computation, comprising: deriving element-level equations using the defined Lagrange shape functions; computing the Jacobian matrix and determinant based on the derived equations; validating the geometric transformations of the modified high-order finite element mesh; and generating symbolic representations of the element stiffness matrices prior to numerical evaluation.

10. The method (300) as claimed in claim 8, wherein the processor (102) utilising the retrieved geometric information, including node coordinates and element connectivity, to generate the boundary node modifications required for subparametric finite element computations.

Description:TECHNICAL FIELD
[0001] The present invention relates generally to the field of finite element analysis (FEA) and more particularly to a system and method for subparametric finite element analysis for high-order meshes in computational domains, with applicability in areas such as structural mechanics, fluid dynamics, electromagnetics, and thermal analysis.

BACKGROUND
[0002] The following description of the related art is intended to provide background information pertaining to the field of disclosure. This section may include certain aspects of the art that may be related to various features of the present disclosure. However, it should be appreciated that this section is used only to enhance the understanding of the reader with respect to the present disclosure, and not as admissions of the prior art.
[0003] Finite Element Method (FEM) is widely recognized as a versatile numerical technique that facilitates the solution of complex engineering and scientific problems by discretizing a continuous domain into smaller elements and approximating field variables within each element. FEM is particularly suited for problems involving complex geometries, heterogeneous materials, and variable boundary conditions, and its accuracy depends significantly on the quality of mesh generation, element formulations, and numerical solution methods employed.
[0004] Over the years, various computational tools have been developed to generate finite element meshes and perform simulations, including popular frameworks such as Gmsh, MFEM, and MATLAB-based higher-order mesh generators. Gmsh, for example, provides unstructured mesh generation using algorithms like Delaunay and Frontal methods, offering robust solutions for simple domains. However, it faces challenges in generating structured boundary layer meshes, maintaining geometric fidelity in curved domains, and handling very large computational problems efficiently. MATLAB-based subparametric higher-order mesh generators allow accurate modeling of curved geometries and complex elements, yet these solutions are often constrained by sequential computations, limited scalability, and high computational times for large-scale problems. Similarly, parallel FEM frameworks and GPU-accelerated solvers improve computation speed but are generally restricted by implementation complexity, reliance on specific programming languages such as C++, and limited flexibility in integrating adaptive mesh refinement techniques.
[0005] These limitations become more pronounced when high-order elements are required to accurately capture geometric features or when adaptive meshing is necessary to optimize the distribution of elements based on local problem characteristics. The challenges include maintaining mesh integrity, accurately mapping reference element coordinates to physical coordinates, ensuring computational efficiency for large-scale problems, and enabling smooth integration with high-performance computing resources. Existing software solutions often encounter issues such as incomplete mesh connectivity, non-uniform elements, and difficulties in handling CAD-imported geometries, high memory consumption, and insufficient support for automated symbolic computations needed for high-fidelity simulations.
[0006] Therefore, there is a need for a solution that can efficiently handle high-order meshes with curved geometries, heterogeneous material distributions, and large computational domains, while providing scalability, numerical stability, and compatibility with parallel computation platforms such as GPU-based systems.

OBJECTS OF THE PRESENT DISCLOSURE
[0007] Some of the objects of the present disclosure, which at least one embodiment herein satisfies are as listed herein below.
[0008] A general object of the present disclosure is to provide a subparametric finite element analysis system that enables accurate representation of complex geometries including curved boundaries and heterogeneous materials while maintaining numerical stability and fidelity across large-scale computational domains.
[0009] An object of the present disclosure is to provide a high-order mesh generation framework that retrieves and organizes node coordinates, element connectivity, and boundary information to support adaptive subparametric meshing for enhanced simulation accuracy.
[0010] Another object of the present disclosure is to provide a system that performs element-level symbolic computations for generating element equations, evaluates Jacobian matrices and determinants for each element, and validates geometric transformations to ensure precise mapping between reference and physical coordinates.
[0011] Yet another object of the present disclosure is to provide a parallelized computation framework that assembles global stiffness matrices, computes element stiffness matrices, and applies boundary conditions using GPU-accelerated techniques to enable efficient and scalable solution of large, high-resolution finite element problems.

SUMMARY
[0012] This section is provided to introduce certain objects and aspects of the present disclosure in a simplified form that are further described below in the detailed description. This summary is not intended to identify the key features or the scope of the claimed subject matter.
[0013] Aspects of the present invention relate generally to the field of finite element analysis (FEA) and more particularly to a system and method for subparametric finite element analysis for high-order meshes in computational domains, addressing challenges of mesh generation, geometric accuracy, parallel computation, and efficient evaluation of element and global stiffness matrices.
[0014] An aspect of the present disclosure provides a subparametric finite element analysis system and method designed for high-order meshes in computational domains with applications in structural mechanics, fluid dynamics, electromagnetics, and thermal analysis. The system provides a framework for efficiently generating high-order finite element meshes, retrieving and organizing essential geometric information including node coordinates, element connectivity, and boundary data, and performing computations necessary for accurate subparametric finite element analysis. The system utilises a processor operatively coupled to memory to execute instructions for mesh generation, modification, and subsequent finite element computations in a scalable and parallelized manner. The system initiates the process by generating a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface, and subsequently retrieves node coordinates, element connectivity, and geometric information required for subparametric finite element computations. The processor modifies the coordinates of boundary nodes of the generated mesh to enable subparametric transformations that ensure accuracy and stability in subsequent computations. Shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh are defined for each node, evaluated at integration points, and stored in structured arrays or data matrices to facilitate element-level computation.
[0015] In addition, element-level governing equations are computed using the defined shape functions, followed by calculation of a Jacobian matrix and its determinant for each finite element based on the computed equations, with the Jacobian serving to validate the geometric transformations of the modified high-order finite element mesh. Element stiffness matrices are further computed using subparametric finite element formulations based on the Jacobian matrix, and assembled into a global stiffness matrix using parallel computation executed on a CUDA-enabled graphics processing unit. Physical boundary conditions are applied to the global stiffness matrix in a parallelized manner on the GPU, resulting in a numerical solution of the finite element problem defined over the computational domain.
[0016] In an aspect, the system is capable of computing the Jacobian matrix and its determinant for each element using subparametric transformations to map reference element coordinates to physical element coordinates accurately.
[0017] In an aspect, symbolic computation is utilized for deriving element-level equations from the defined Lagrange shape functions, computing the Jacobian and its determinant, validating geometric transformations, and generating symbolic representations of element stiffness matrices prior to numerical evaluation.
[0018] In an aspect, the processor further supports adaptive refinement of the high-order finite element mesh based on geometric features or element-specific parameters extracted from the mesh data, ensuring improved accuracy and efficiency in complex domains.
[0019] In an aspect, computations of element stiffness matrices are performed in parallel across multiple finite elements, and assembly of the global stiffness matrix along with the application of boundary conditions are conducted in parallel on the CUDA-enabled GPU to enhance scalability and reduce execution time.
[0020] In an aspect, the processor exploits the retrieved geometric information, including node coordinates and element connectivity, to define and modify boundary nodes for subparametric finite element computations.
[0021] In an aspect, shape functions corresponding to the selected polynomial order are systematically defined, evaluated at integration points, and stored in structured arrays or data matrices, facilitating precise and efficient element-level computation.
[0022] Another aspect of the proposed disclosure pertains to a method comprising computational steps executed by a processor, including high-order mesh generation, retrieval of node and geometric data, boundary node modification, definition and evaluation of Lagrange shape functions for each finite element, computation of element-level governing equations, calculation and validation of the Jacobian matrix and determinant, computation of element stiffness matrices using subparametric formulations, assembly of the global stiffness matrix in parallel on a CUDA-enabled GPU, and application of physical boundary conditions. The method also incorporates symbolic computation for deriving element-level equations, validating geometric transformations, and generating symbolic representations of stiffness matrices prior to numerical evaluation.
[0023] Various objects, features, aspects and advantages of the inventive subject matter will become more apparent from the following detailed description of preferred embodiments, along with the accompanying drawing figures in which like numerals represent like components.

BRIEF DESCRIPTION OF DRAWINGS
[0024] The accompanying drawings are included to provide a further understanding of the present disclosure and are incorporated in and constitute a part of this specification. The drawings illustrate exemplary embodiments of the present disclosure and, together with the description, serve to explain the principles of the present disclosure. The diagrams are for illustration only, which thus is not a limitation of the present disclosure.
[0025] FIG. 1 illustrates an exemplary block diagram of a subparametric finite element analysis system, in accordance with an embodiment of the present disclosure.
[0026] FIG. 2 illustrates an exemplary block diagram of computational framework of proposed subparametric finite element analysis system, in accordance with an embodiment of the present disclosure.
[0027] FIG. 3 illustrates an exemplary flow diagram depicting a method for performing subparametric finite element analysis, in accordance with an embodiment of the present disclosure.
[0028] FIG. 4 illustrates a block diagram of an example computer system in which or with which embodiments of the present disclosure may be implemented, in accordance with an embodiment of the present disclosure.
[0029] FIG. 5 illustrates an exemplary view of Gmsh Waveguide mesh with boundary nodes modified using subparametric transformations, in accordance with an embodiment of the present disclosure.

DETAILED DESCRIPTION
[0030] The ensuing description provides exemplary embodiments only, and is not intended to limit the scope, applicability, or configuration of the disclosure. Rather, the ensuing description of the exemplary embodiments will provide those skilled in the art with an enabling description for implementing an exemplary embodiment. It should be understood that various changes may be made in the function and arrangement of elements without departing from the spirit and scope of the disclosure as set forth.
Definitions:
Subparametric Finite Element Analysis (sFEA): A computational technique that employs high-order finite element meshes and subparametric transformations to solve partial differential equations over complex geometries while enabling accurate mapping between reference and physical element coordinates.
High-Order Finite Element Mesh: A discretization of a computational domain into finite elements where nodes can exist along edges and faces, and polynomial interpolation of higher degree is used to accurately capture curved boundaries and heterogeneous material distributions.
Shape Functions: Mathematical functions associated with each node of a finite element that interpolate field variables within the element and enable computation of element-level equations for numerical analysis.
Element-Level Governing Equations: Algebraic equations derived from the discretization of partial differential equations over a finite element using defined shape functions, representing the behavior of field variables at the element level.
Jacobian Matrix: A matrix representing the transformation between reference element coordinates and physical element coordinates, used to evaluate geometric consistency and validate subparametric transformations within each finite element.
Element Stiffness Matrix: A numerical matrix computed for each finite element based on element-level governing equations and Jacobian transformations, representing the resistance of the element to deformation or variation in field variables.
Global Stiffness Matrix: An assembled matrix that integrates all element stiffness matrices of a finite element mesh to represent the behavior of the entire computational domain and facilitate numerical solution of field equations.
CUDA-enabled Graphics Processing Unit (GPU): A parallel computing device capable of executing multiple numerical computations simultaneously to accelerate the evaluation of element stiffness matrices, assembly of global stiffness matrices, and application of boundary conditions in finite element analysis.
[0031] Embodiments of the present invention relate generally to the field of finite element analysis (FEA) and more particularly to a system and method for subparametric finite element analysis for high-order meshes in computational domains, addressing challenges of mesh generation, geometric accuracy, parallel computation, and efficient evaluation of element and global stiffness matrices.
[0032] Various embodiments of the present disclosure are described using FIGs. 1 to 5.
[0033] Referring to FIG. 1, an exemplary block diagram of a subparametric finite element analysis (sFEA) system (100), interchangeably referred to as system (100), is illustrated. The proposed system (100) can be implemented as a software-based platform operable through a computing interface including, but not limited to, a desktop application, web-based application, mobile application, or integrated computational environment configured to perform finite element analysis over complex computational domains. The system (100) includes a processor (102), a memory (104), and one or more interfaces (108). The system (100) may operate as a software platform, application, or computing service and may be communicatively coupled to external devices or servers through a communication network.
[0034] The communication network may include at least one of a wired network or a wireless network including Wi-Fi, cellular communication networks, Bluetooth communication, satellite communication networks, or equivalent communication technologies.
[0035] The processor (102) may be implemented as one or more microprocessors, microcontrollers, digital signal processors, central processing units, or logic circuitries capable of executing computer-readable instructions. The processor (102) is configured to fetch and execute instructions stored in the memory (104) to perform subparametric finite element analysis operations including high-order mesh processing, geometric transformation, shape function definition, element-level computation, and global system assembly.
[0036] The memory (104) stores computer-readable instructions, data structures, and computational parameters required for execution of finite element analysis workflows. The memory (104) may include non-transitory storage such as random-access memory, read-only memory, flash memory, or other suitable storage media. The stored instructions, when executed by the processor (102), enable execution of mesh generation, symbolic computation, numerical evaluation, and parallel processing tasks associated with subparametric finite element analysis.
[0037] The interface(s) (108) include input and output interfaces configured to facilitate interaction between the system (100) and external devices, computing platforms, or visualization environments. The interface(s) (108) further enable communication between internal components of the system (100), including a processing engine(s) (110) and a database (112). The database (112) stores mesh data, geometric information, shape function data, symbolic expressions, numerical matrices, and intermediate computational results generated during execution of the finite element analysis.
[0038] In an embodiment, the processing engine(s) (110) are implemented as a combination of hardware and processor-executable instructions stored on a non-transitory machine-readable storage medium. The processing engine(s) (110) may alternatively be implemented using electronic circuitry configured to perform dedicated computational operations. The processing engine(s) (110) include a mesh generation and extraction module (114), a boundary geometry transformation module (116), a shape function definition module (118), a symbolic and element equation computation module (120), a parallel element and global assembly module (122), an adaptive mesh refinement module (124), and other module(s) (126). The other module(s) (128) perform supplementary operations that support execution, optimization, or management of subparametric finite element analysis within the system (100).
[0039] In an embodiment, the mesh generation and extraction module (114) generates a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface. The Mesh Generation and Extraction Module (114) further extracts, from the generated high-order finite element mesh, node coordinates, element connectivity, and geometric information required for execution of subparametric finite element computations.
[0040] In an embodiment, the boundary geometry transformation module (116) modifies coordinates of boundary nodes of the generated high-order finite element mesh to enable subparametric geometric transformations by selectively updating only the geometric representation of boundary elements while maintaining the interpolation order of the solution field. The boundary geometry transformation module (116) identifies boundary nodes using the extracted element connectivity and topological information and classifies the nodes as corner nodes, mid-side nodes, or higher-order boundary nodes associated with curved edges or surfaces.
[0041] The boundary geometry transformation module (116) utilises the retrieved geometric information, including node coordinates and element connectivity, to compute updated physical positions for the identified boundary nodes based on prescribed geometric constraints, analytical boundary descriptions, or externally defined geometric mappings. The updated boundary node coordinates reflect the true physical geometry of the computational domain, while interior node coordinates remain unchanged, thereby enforcing a subparametric formulation in which the geometry approximation order is lower than the field interpolation order.
[0042] In one example, for a high-order triangular finite element approximating a curved boundary, the boundary geometry transformation module (116) repositions mid-side boundary nodes along an exact circular or spline-defined boundary curve derived from geometric input data, while preserving the original locations of interior nodes. This ensures that the curved boundary is accurately represented without increasing the polynomial order of the geometric mapping for interior regions of the mesh.
[0043] In another example, for a two-dimensional high-order finite element mesh representing a complex surface, the boundary geometry transformation module (116) projects boundary nodes onto a surface using parametric surface equations or interpolation of discrete surface points, thereby correcting geometric distortion introduced during initial mesh generation. The resulting boundary-modified mesh supports accurate Jacobian computation and stable numerical integration during subsequent subparametric finite element computations.
[0044] In an embodiment, the shape function definition module (118) generates shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh by constructing interpolation functions that relate nodal degrees of freedom to field variable values within each finite element. The shape function definition module (118) first determines the polynomial order assigned to each finite element based on analysis requirements or mesh configuration parameters associated with the modified high-order finite element mesh. The shape function definition module (118) defines shape functions for each node of the modified high-order finite element by formulating Lagrange interpolation polynomials over a reference element coordinate system. The formulation associates each nodal location with a unique shape function such that the shape function evaluates to unity at its corresponding node and to zero at all other nodes, thereby ensuring nodal interpolation consistency across the element.
[0045] This shape function definition module (118) evaluates the defined shape functions at predetermined integration points within the reference element using numerical integration schemes appropriate to the selected polynomial order. The evaluated shape function values and their derivatives are computed in the reference coordinate space and subsequently mapped to the physical element using the modified geometric representation of the high-order finite element mesh. Further, this shape function definition module (118) stores the evaluated shape functions and associated derivative values in structured arrays or data matrices indexed by element, node, and integration point. These stored representations enable efficient reuse during element-level governing equation formulation, Jacobian computation, and stiffness matrix evaluation without redundant recalculation.
[0046] In one example, for a curved high-order triangular finite element with a selected polynomial order greater than one, the shape function definition module (118) constructs a set of Lagrange shape functions over a standard triangular reference element. The shape functions are evaluated at gaussian integration points and stored in element-specific matrices, which are subsequently accessed during numerical integration of element stiffness contributions while maintaining consistency with the modified boundary geometry.
[0047] In an embodiment, the symbolic and element equation computation module (120) computes element-level governing equations using the generated shape functions by expressing field variables, gradients, and constitutive relationships in symbolic form over each finite element. The symbolic and element equation computation module (120) formulates symbolic expressions that relate nodal degrees of freedom to physical quantities within the element domain, thereby establishing a symbolic representation of the governing equations prior to numerical evaluation. This derives element-level equations by combining the Lagrange shape functions associated with each finite element with material properties and governing physical relations. The derivation is performed in a reference coordinate space, enabling analytical representation of interpolation functions, spatial derivatives, and integral expressions required for finite element formulation.
[0048] In addition, this symbolic and element equation computation module (120) computes, for each finite element, a Jacobian matrix and a determinant of the Jacobian matrix by symbolically differentiating the geometric mapping between reference element coordinates and physical element coordinates. The computed Jacobian matrix represents the local geometric transformation, while the determinant of the Jacobian quantifies element distortion and scaling introduced by boundary modifications.
[0049] Further, this symbolic and element equation computation module (120) validates the geometric transformations of the high-order finite element mesh by analyzing the computed Jacobian matrix and determinant for admissibility conditions associated with subparametric finite element analysis. This validation ensures consistency between the modified boundary geometry and the interior element formulation without altering the polynomial order of the solution approximation.
[0050] Furthermore, this symbolic and element equation computation module (120) generates symbolic representations of element stiffness matrices by symbolically integrating products of shape functions, their derivatives, and material coefficients over the reference element domain. These symbolic stiffness matrix expressions are preserved prior to numerical quadrature and global assembly, enabling efficient evaluation across multiple elements while maintaining formulation consistency.
[0051] In one example, for a curved high-order triangular element, the symbolic and element equation computation module (120) symbolically expresses displacement fields using Lagrange shape functions, derives strain–displacement relations, and computes the Jacobian matrix representing the curved geometry mapping. The resulting symbolic stiffness matrix formulation is stored and later numerically evaluated during parallel assembly while ensuring the geometric transformation remains valid throughout the analysis.
[0052] In an embodiment, the parallel element and global assembly module (122) computes element stiffness matrices using subparametric finite element formulations by numerically evaluating the previously derived element-level equations in the physical coordinate space defined by each finite element. The parallel element and global assembly module (122) apply subparametric transformations to map reference element coordinates to physical element coordinates using the Jacobian matrix and the determinant of the Jacobian matrix associated with each finite element. This utilizes the Jacobian matrix to transform shape function derivatives from the reference coordinate system to the physical coordinate system, thereby enabling accurate numerical integration of stiffness contributions within each element. The determinant of the Jacobian matrix is applied as a scaling factor during numerical integration to account for geometric distortion introduced by curved boundaries or boundary node modifications.
[0053] The parallel element and global assembly module (122) distribute element-level stiffness matrix computations across multiple processing threads on the graphics processing unit (130). Each processing thread independently evaluates stiffness contributions for one or more finite elements, allowing concurrent computation across the high-order finite element mesh while maintaining consistency of subparametric formulations. Further, assembles a global stiffness matrix by mapping local element stiffness matrices into a global matrix structure based on element connectivity and global degree-of-freedom indexing. The assembly operation is executed in parallel on the graphics processing unit (130) using concurrent accumulation strategies that ensure correct aggregation of element contributions at shared nodes.
[0054] Further, the parallel element and global assembly module (122) applies physical boundary conditions to the assembled global stiffness matrix by modifying selected matrix entries and load vectors in accordance with prescribed constraints associated with the computational domain. Boundary condition enforcement is performed in parallel on a CUDA-enabled graphics processing unit (130), enabling simultaneous processing of multiple constrained degrees of freedom without serial bottlenecks.
[0055] Furthermore, this parallel element and global assembly module (122) produces a numerical solution of the finite element problem by completing the global matrix formulation after boundary condition application, thereby preparing the system for solution using compatible numerical solvers while preserving the geometric and formulation integrity of the subparametric finite element analysis.
[0056] In one example, a large high-order mesh containing curved boundary elements is partitioned across processing threads of the graphics processing unit (130), where each thread computes stiffness matrices using element-specific Jacobian matrices. The resulting stiffness matrices are assembled concurrently into a global stiffness matrix, and displacement constraints along a curved boundary are applied in parallel, enabling efficient solution of a large-scale structural analysis problem.
[0057] In an embodiment, the adaptive mesh refinement module (124) performs adaptive refinement of the high-order finite element mesh by analysing geometric features and element-specific parameters derived from node coordinates, element connectivity, and geometric information extracted from the generated high-order finite element mesh. This adaptive mesh refinement module (124) identifies finite elements requiring refinement by evaluating local geometric characteristics such as curvature, boundary proximity, element distortion, and variation in element size across the computational domain.
[0058] Further, determines refinement regions by comparing element-specific parameters against predefined refinement criteria stored in the memory (104), including tolerance thresholds associated with geometric complexity or solution sensitivity indicators. Elements satisfying the refinement criteria are marked for subdivision or polynomial order adjustment while preserving mesh conformity and continuity across adjacent elements. Further, updates node coordinates and element connectivity information for refined elements by introducing additional nodes or modifying polynomial order assignments, while maintaining compatibility with subparametric finite element formulations. The refinement process ensures that boundary representations remain consistent with existing geometric transformations applied to boundary nodes, thereby preserving the validity of Jacobian matrices and subparametric mappings.
[0059] Furthermore, this adaptive mesh refinement module (124) integrates the refined mesh data into subsequent computational stages by updating geometric information accessible to the shape function definition module (118), symbolic and element equation computation module (120), and parallel element and global assembly module (122). This integration allows refined regions to participate seamlessly in element-level computation and global matrix assembly without disrupting the finite element workflow.
[0060] In one example, a computational domain containing a curved boundary region exhibits high geometric curvature relative to interior regions. The adaptive mesh refinement module (124) identifies boundary-adjacent elements with elevated curvature parameters and refines only those elements by increasing local resolution, while leaving interior elements unchanged, thereby improving numerical accuracy near the boundary without increasing computational cost across the entire mesh.
[0061] Referring to FIG. 2, an exemplary block diagram (200) illustrating a computational framework of the proposed sFEA system (100) is disclosed. The diagram illustrates interaction among functional modules of the system (100) as previously described with reference to FIG. 1, and collectively depicts execution of subparametric finite element analysis across a computational domain.
[0062] At a top level, the sFEA framework functions as a central coordination layer that manages operations ranging from mesh preparation to numerical solution generation. The framework integrates a mesh handling branch and a subparametric solver branch, thereby maintaining logical separation between mesh-related operations and finite element computations while enabling structured data exchange between the branches.
[0063] On a mesh handling side, a Gmsh API interfaces directly with a Python-based application programming interface of Gmsh to generate an initial finite element mesh and to access geometric and topological data associated with the computational domain. A mesh module builds upon the generated mesh to produce high-order finite element representations suitable for curved boundaries and complex geometries. The high-order representation supports subparametric formulations by enabling accurate geometric approximation without increasing polynomial order of solution fields.
[0064] The processor (102) retrieves node coordinates, element connectivity, and associated attributes from the mesh data structures and organizes the retrieved information into array-based or matrix-based formats suitable for symbolic and numerical processing. In addition, the processor (102) subsequently adjusts boundary node coordinates to support subparametric geometric transformations, thereby ensuring consistency between boundary representation and element formulation in downstream computations.
[0065] On a solver side, the processor (102) operates as a computational core responsible for finite element calculations. Within the solver, the processor (102) evaluates shape functions corresponding to the polynomial order of the high-order mesh. These shape functions define interpolation of field variables within each finite element and serve as a basis for formulation of element-level equations.
[0066] Further, the processor (102) computes discrete element-level governing equations using the evaluated shape functions, translating physical governing relations into algebraic expressions at the element scale, and computes Jacobian matrices and corresponding determinants for each finite element, enabling coordinate mapping between reference and physical domains and validating correctness of subparametric geometric transformations. Computed element equations and Jacobian information are executed on the graphics processing unit (130). Element stiffness matrices are computed by the processor (102) in parallel, thereby accelerating finite element assembly for large meshes. The processor (102) also executed on the graphics processing unit (130), incorporates physical boundary conditions into the assembled system by modifying stiffness matrix entries and associated vectors to enforce prescribed constraints.
[0067] In an exemplary embodiment, the proposed system (100) (sFEA) is applicable across diverse engineering and biomedical domains where accurate modelling of complex geometries and multiphysics interactions is required. In structural and mechanical engineering, the framework supports stress, strain, and deformation analysis of components such as aircraft structures, automotive parts, and load-bearing systems. In energy and electronics applications, sFEA enables simulation of electromagnetic wave propagation in waveguides, thermal behaviour in microelectronic devices, and coupled fluid–thermal phenomena in pipelines. The use of high-order finite elements and subparametric geometric transformations allows improved accuracy in regions involving curved boundaries and steep field gradients compared to conventional finite element approaches.
[0068] In an exemplary embodiment, the proposed system (100) (sFEA) further extends to biomedical and multiphysics applications, including ocular biomechanics and aqueous humor dynamics influenced by thermal, fluidic, and magnetic fields. The framework enables precise discretization of thin, curved anatomical geometries such as the anterior chamber of the eye and supports coupled analyses relevant to intraocular pressure regulation and targeted drug delivery. By integrating Python-based mesh generation with GPU-accelerated parallel computation, sFEA overcomes scalability limitations of traditional sequential finite element implementations. This enables efficient high-resolution simulations and parametric studies across large computational domains, making the framework suitable for advanced electromagnetic waveguide analysis as well as complex biomedical and engineering systems requiring high accuracy and computational efficiency.
[0069] In an exemplary implementation, the proposed system (100) operates in near real-time by dynamically processing high-order finite element meshes and associated geometric data as inputs are updated. The processor (102) performs on-the-fly modification of boundary node coordinates, evaluates shape functions, and computes element-level equations while concurrently executing parallel stiffness matrix computation and assembly on graphics processing unit (130). The system (100) continuously applies boundary conditions and updates the numerical solution in response to changes in geometry or physical parameters, thereby enabling interactive analysis of complex computational domains with minimal latency.
[0070] Referring to FIG. 3, an exemplary flowchart illustrates the working of a proposed method (300) for performing subparametric finite element analysis. The method begins at block (302), where a processor (102) generates a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface. The processor controls mesh generation parameters to create finite elements of a selected polynomial order, enabling accurate representation of complex geometries and curved boundaries required for subparametric finite element analysis.
[0071] Continuing further, at block (304), the processor (102) retrieves, from the generated high-order finite element mesh, node coordinates, element connectivity, and geometric information required for subparametric finite element computations. The processor (102) organizes the retrieved information into structured data representations that define element topology, nodal relationships, and boundary characteristics, enabling consistent access during subsequent computational stages.
[0072] Continuing further, at block (306), the processor (102) modifies coordinates of boundary nodes of the generated high-order finite element mesh to enable geometric transformations. The processor (102) utilises the retrieved geometric information, including node coordinates and element connectivity, to determine boundary nodes and apply coordinate modifications that support subparametric transformations while preserving mesh continuity and geometric validity.
[0073] Continuing further, at block (308), the processor (102) defines shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh. The processor selects a polynomial order for each finite element, defines Lagrange shape functions for each node of the modified high-order finite element, evaluates the shape functions at each integration point of the finite element, and stores the evaluated shape functions in structured arrays or data matrices for element-level computation.
[0074] Continuing further, at block (310), the processor (102) computes element-level governing equations using the defined shape functions. The processor performs symbolic computation using Symengine’s API by deriving element-level equations using the defined Lagrange shape functions, computing a Jacobian matrix and a determinant of the Jacobian based on the derived equations, validating the geometric transformations of the modified high-order finite element mesh, and generating symbolic representations of element stiffness matrices prior to numerical evaluation.
[0075] Continuing further, at block (312), the processor (102) computes, for each finite element, a Jacobian matrix and a determinant of the Jacobian matrix based on the computed element-level governing equations. The processor (102) uses the computed Jacobian matrix to map reference element coordinates to physical element coordinates and to validate the correctness of the applied geometric transformations.
[0076] Continuing further, at block (314), the processor (102) computes element stiffness matrices using subparametric finite element formulations based on the computed Jacobian matrix. The stiffness computation incorporates the validated geometric transformations and shape function evaluations to ensure numerical consistency at the element level.
[0077] Continuing further, at block (316), the processor (102) assembles a global stiffness matrix from the element stiffness matrices using parallel computation executed on a CUDA-enabled graphics processing unit (130). The parallel assembly using CUDA or Numba enables simultaneous processing of stiffness contributions from multiple finite elements, improving computational efficiency for large-scale meshes.
[0078] Continuing further, at block (318), the processor (102) applies physical boundary conditions to the global stiffness matrix using parallel computation executed on the graphics processing unit (130) to produce a numerical solution of the finite element problem defined on the computational domain. The applied boundary conditions enforce physical constraints and complete the formulation of equations for numerical solution.
[0079] Referring to FIG. 4, an example of an embedded computer system (400) may include an external storage device (410), a bus (420), a main memory (430), a read-only memory (440), a mass storage device (450), communication port(s) (460), and a processor (470). Those skilled in the art will appreciate that the computer system (400) may include more than one processor and communication ports. The processor (470) may include various modules associated with embodiments of the present disclosure. The communication port(s) (460) may be any of an RS-232 port for use with a modem-based dialup connection, a 10/100 Ethernet port, a Gigabit or 10 Gigabit port using copper or fibre, a serial port, a parallel port, or other existing or future ports. The communication port(s) (460) may be chosen depending on a network, such as a Local Area Network (LAN), Wide Area Network (WAN), or any network to which the computer system (400) connects. The main memory (430) may be random access memory (RAM), or any other dynamic storage device commonly known in the art. The read-only memory (440) may be any static storage device(s) including, but not limited to, a Programmable Read Only Memory (PROM) chips for storing static information e.g., start-up or basic input/output system (BIOS) instructions for the processor (470). The mass storage device (450) may be any current or future mass storage solution, which may be used to store information and/or instructions.
[0080] The bus (420) communicatively couples the processor (470) with the other memory, storage, and communication blocks. The bus (420) can be, e.g., a Peripheral Component Interconnect (PCI) / PCI Extended (PCI-X) bus, Small Computer System Interface (SCSI), universal serial bus (USB), or the like, for connecting expansion cards, drives, and other subsystems as well as other buses, such a front side bus (FSB), which connects the processor (470) to the computer system (400).
[0081] Optionally, operator and administrative interfaces, e.g. a display, keyboard, and a cursor control device, may also be coupled to the bus (420) to support direct operator interaction with the computer system (400). Other operator and administrative interfaces may be provided through network connections connected through the communication port(s) (460). In no way should the aforementioned exemplary computer system (400) limit the scope of the present disclosure.
[0082] Thus, the proposed system (100) and method (300) provide an efficient framework for performing subparametric finite element analysis using high-order meshes with controlled geometric transformations. The disclosed framework enables accurate handling of complex geometries, improves numerical stability through validated Jacobian computations, and supports scalable computation via parallel processing on a graphics processing unit. Thereby enhances computational efficiency and solution accuracy for large-scale finite element problems.
[0083] While considerable emphasis has been placed herein on the preferred embodiments, it will be appreciated that many embodiments can be made and that many changes can be made in the preferred embodiments without departing from the principles of the disclosure. These and other changes in the preferred embodiments of the disclosure will be apparent to those skilled in the art from the disclosure herein, whereby it is to be distinctly understood that the foregoing descriptive matter is to be interpreted merely as illustrative of the disclosure and not as a limitation.

Experimental Analysis
[0084] The performance of the prototype is evaluated through comparisons with results from MATLAB-based implementations. The outcomes of these experiments are presented in Tables 1-4, which correspond directly to mesh shown in FIG. 5.
[0085] Table 1 provides computed TM cut-off frequencies across increasing polynomial orders of elements. The close agreement between sFEA and MATLAB results demonstrates that the framework maintains accuracy while extending the solution to higher-order elements. For better performance, CUDA-enabled GPUs are used for parallel processing so that several computations can be performed in parallel on different elements simultaneously. CUDA really accelerates operations like computing the element stiffness matrices and assembling the global stiffness matrix. Table 2 shows the time comparison between using different backends for symbolic computation. Table 3 presents the execution times for different configurations, and the results clearly indicate that parallel processsing dramatically improves performance, keeping solution times low even as the order of the elements increases. Parallelization based on a GPU dramatically reduces execution time, especially for large-scale problems with a large number of elements. Table 4 shows time comparison between sequential / Numba / CUDA (GPU).

Table 1: TM cut-off frequencies across increasing polynomial orders of elements

MATLAB order=1 order=2 order=3 order=4 order=5
TM 4.498 4.498 4.496 4.498 4.497 4.498
4.559 4.56 4.557 4.559 4.558 4.559
6.398 6.401 6.397 6.398 6.398 6.398
6.399 6.402 6.397 6.399 6.399 6.4
7.096 7.1 7.092 7.096 7.097 7.096
NO. OF ELEMENTS 20001 5156 1508 939 408
NODES 10292 10607 7022 7765 5309
TIME (SECONDS) (ONLY FOR TM) 14980 20076 8831 7490 2948
MESH SIZE 0.017 0.033 0.06 0.075 0.15
Table 2: Element Stiffness Matrix Computations
MESH SIZE (GMSH PYTHON) 0.008
NO. OF ELEMENTS 93996
NODES 47627
ORDER 1
SYMENGINE: COMPUTATIONAL TIME IN SECONDS 320.89
SYMPY: COMPUTATIONAL TIME IN SECONDS 23115.39

Table 3: Element Stiffness Matrix Computations
MESH SIZE (GMSH PYTHON) 0.15
NO. OF ELEMENTS 352
NODES 4579
ORDER 5
SYMENGINE: COMPUTATIONAL TIME IN SECONDS 1075.25
SYMENGINE + JOBLIB (2 JOBS): COMPUTATIONAL TIME IN SECONDS 978.06

Table 1: Parallel Assembly of element stiffness matrices using different methods
MESH SIZE (GMSH PYTHON) 0.005 0.008 0.01
NO. OF ELEMENTS 239333 93996 60254
NODES 120672 47627 30630
ORDER 1 1 1
SEQUENTIAL ASSEMBLY: COMPUTATIONAL TIME IN SECONDS 12.033 4.877 2.837
NUMBA: COMPUTATIONAL TIME IN SECONDS 5.693 3.613 2.389

[0086] The proposed system (100) or sFEA has been experimentally validated using benchmark problems, and the obtained results demonstrate strong agreement with reference solutions. The experimental evaluation confirms that the integration of higher-order subparametric formulations, efficient symbolic computation, and CUDA-enabled parallel processing provides accurate numerical results while significantly reducing computational time and memory requirements. The observed convergence behaviour, reduced element count, and accelerated assembly and solution processes collectively indicate that the proposed methodology effectively overcomes the scalability and performance limitations of conventional finite element tools, even for complex geometries and large-scale computational domains.

ADVANTAGES OF THE PRESENT DISCLOSURE
[0087] The present disclosure provides accurate and high-fidelity generation of high-order finite element meshes for complex computational domains, ensuring that curved boundaries and heterogeneous material distributions are precisely represented, thereby improving the reliability of numerical simulations across structural, fluid, electromagnetic, and thermal analysis applications.
[0088] The present disclosure provides efficient computation of element and global stiffness matrices significantly reducing execution time for large-scale, high-resolution problems while maintaining numerical stability and computational consistency.
[0089] The present disclosure provides precise validation of geometric transformations and adaptive refinement of high-order finite element meshes, enhancing solution accuracy in regions with complex geometry or high field gradients without unnecessarily increasing computational load for simpler regions.
[0090] The present disclosure provides a unified, Python-based framework that systematically defines and evaluates shape functions corresponding to selected polynomial orders for each modified high-order finite element, ensuring accurate interpolation at integration points, improved convergence rates, and scalability to large and complex computational domains with millions of degrees of freedom.
, Claims:1. A subparametric finite element analysis system (100) comprising:
a processor (102);
a memory (104) operatively coupled to the processor (102), wherein the processor (102) is configured to execute instructions stored in the memory to:
generate a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface;
retrieve from the generated high-order finite element mesh, node coordinates, element connectivity, and geometric information required for subparametric finite element computations;
modify coordinates of boundary nodes of the generated high-order finite element mesh to enable subparametric transformations;
generate shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh;
compute element-level governing equations using the generated shape functions;
compute, for each finite element, a Jacobian matrix and a determinant of the Jacobian matrix based on the computed element-level governing equations, the computed Jacobian matrix being used to validate the geometric transformations;
compute element stiffness matrices using subparametric finite element formulations based on the computed Jacobian matrix;
assemble a global stiffness matrix from the element stiffness matrices using parallel computation executed on a CUDA-enabled graphics processing unit (130) operatively coupled to the processor (102); and
apply physical boundary conditions to the global stiffness matrix using parallel computation executed on the CUDA-enabled graphics processing unit (130) to produce a numerical solution of a finite element problem defined on the computational domain.
2. The system (100) as claimed in claim 1, wherein the Jacobian matrix and determinant of the Jacobian matrix are computed for each element using subparametric transformations to map reference element coordinates to physical element coordinates.
3. The system (100) as claimed in claim 1, wherein the processor (102) is further configured to perform symbolic computation to compute the element-level governing equations, wherein to perform the symbolic computation, the processor is configured to:
derive element-level equations using Lagrange shape functions generated for each finite element;
compute the Jacobian matrix and determinant of the Jacobian for each element based on the derived element-level equations;
validate the geometric transformations of the high-order finite element mesh using the computed Jacobian and determinant; and
generate symbolic representations of the element stiffness matrices prior to numerical evaluation and assembly into the global stiffness matrix.
4. The system (100) as claimed in claim 1, wherein the processor (102) is further configured to perform adaptive refinement of the high-order finite element mesh based on geometric features or element-specific parameters received from the node coordinates, element connectivity, and geometric information extracted from the generated high-order finite element mesh.
5. The system (100) as claimed in claim 1, wherein element stiffness matrices are computed in parallel across multiple finite elements on the CUDA-enabled graphics processing unit (130).
6. The system (100) as claimed in claim 1, wherein the processor (102) utilises the retrieved geometric information, including node coordinates and element connectivity, to generate the boundary node modifications required for subparametric finite element computations.
7. The system (100) as claimed in claim 1, wherein the processor (102) is configured to define the shape functions corresponding to the selected polynomial order of the modified high-order finite element mesh, wherein the processor (102):
selects a polynomial order for the finite element;
defines the shape functions for each node of the modified high-order finite element;
evaluates the shape functions at each integration point of the finite element; and
stores the evaluated shape functions in structured arrays or data matrices for element-level computation.
8. A method (300) for performing subparametric finite element analysis, comprising the steps of:
generating (302), by a processor, a high-order finite element mesh of a computational domain using a Gmsh Python application programming interface;
retrieving (304), by the processor, from the generated high-order finite element mesh node coordinates, element connectivity, and geometric information required for subparametric finite element computations;
modifying (306), by the processor, coordinates of boundary nodes of the generated high-order finite element mesh to enable geometric transformations;
defining (308), by the processor, shape functions corresponding to a selected polynomial order of the modified high-order finite element mesh, the step comprising:
selecting a polynomial order for each finite element,
defining Lagrange shape functions for each node of the modified high-order finite element,
evaluating the shape functions at each integration point of the finite element, and
storing the evaluated shape functions in structured arrays or data matrices;
computing (310), by the processor, element-level governing equations using the defined shape functions;
computing (312), by the processor, for each finite element, a Jacobian matrix and a determinant of the Jacobian matrix based on the computed element- equations using Symengine’s API, the Jacobian matrix being used to validate the geometric transformations;
computing (314), by the processor, element stiffness matrices using subparametric finite element formulations based on the computed Jacobian matrix;
assembling (316), by the processor, a global stiffness matrix from the element stiffness matrices using parallel computation executed on a CUDA-enabled graphics processing unit; and
applying (318), by the processor, physical boundary conditions to the global stiffness matrix using parallel computation executed on the CUDA-enabled graphics processing unit to produce a numerical solution of the finite element problem defined on the computational domain.
9. The method as claimed in claim 8, wherein computing the element-level governing equations comprises performing symbolic computation, comprising:
deriving element-level equations using the defined Lagrange shape functions;
computing the Jacobian matrix and determinant based on the derived equations;
validating the geometric transformations of the modified high-order finite element mesh; and
generating symbolic representations of the element stiffness matrices prior to numerical evaluation.
10. The method (300) as claimed in claim 8, wherein the processor (102) utilising the retrieved geometric information, including node coordinates and element connectivity, to generate the boundary node modifications required for subparametric finite element computations.