GPU-based matrix-free finite element solver exploiting symmetry of elemental matrices

Kiran, Utpal; Gautam, Sachin S.; Sharma, Deepak

doi:10.1007/s00607-020-00827-4

Cited by 23 publications

(11 citation statements)

References 38 publications

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“…It was used by Zegard and Paulino (2013) before but implemented for 2D meshes and used a Cholesky-based FEA solver. The symmetry idea was also explored by Duarte et al (2015) for polygonal meshes and by Kiran et al (2020) for FEA of 2D unstructured meshes. The strategy proposed in this paper is more general in terms of its applicability to wide range of finite elements.…”

Section: Ebe Strategymentioning

confidence: 99%

“…The strategy proposed in this paper is more general in terms of its applicability to wide range of finite elements. Unlike Kiran et al (2020) the proposed strategy addresses the implementational challenge associated with 3D finite elements and presents data structure which can efficiently handle large elemental matrices related with 3D elements. The proposed strategy also has no inter thread dependency.…”

Section: Ebe Strategymentioning

confidence: 99%

“…GPU has been utilized to accelerate various computational tasks of FEA, such as computation of elemental matrices Sharma, 2018, 2020), assembling the global stiffness matrix Sharma, 2017, 2019;Kiran et al, 2019) and solution of a system of linear equations (Kiran et al, 2019(Kiran et al, , 2020. Both direct and iterative solvers have been used for solving the linear system of equations on GPU.…”

Section: Introductionmentioning

confidence: 99%

“…, 2019) and solution of a system of linear equations (Kiran et al. , 2019, 2020). Both direct and iterative solvers have been used for solving the linear system of equations on GPU.…”

Section: Introductionmentioning

confidence: 99%

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Acceleration of structural topology optimization using symmetric element-by-element strategy for unstructured meshes on GPU

Ratnakar

Kiran²,

Sharma³

2022

Self Cite

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PurposeStructural topology optimization is computationally expensive due to the involvement of high-resolution mesh and repetitive use of finite element analysis (FEA) for computing the structural response. Since FEA consumes most of the computational time in each optimization iteration, a novel GPU-based parallel strategy for FEA is presented and applied to the large-scale structural topology optimization of 3D continuum structures.Design/methodology/approachA matrix-free solver based on preconditioned conjugate gradient (PCG) method is proposed to minimize the computational time associated with solution of linear system of equations in FEA. The proposed solver uses an innovative strategy to utilize only symmetric half of elemental stiffness matrices for implementation of the element-by-element matrix-free solver on GPU.FindingsUsing solid isotropic material with penalization (SIMP) method, the proposed matrix-free solver is tested over three 3D structural optimization problems that are discretized using all hexahedral structured and unstructured meshes. Results show that the proposed strategy demonstrates 3.1× –3.3× speedup for the FEA solver stage and overall speedup of 2.9× –3.3× over the standard element-by-element strategy on the GPU. Moreover, the proposed strategy requires almost 1.8× less GPU memory than the standard element-by-element strategy.Originality/valueThe proposed GPU-based matrix-free element-by-element solver takes a more general approach to the symmetry concept than previous works. It stores only symmetric half of the elemental matrices in memory and performs matrix-free sparse matrix-vector multiplication (SpMV) without any inter-thread communication. A customized data storage format is also proposed to store and access only symmetric half of elemental stiffness matrices for coalesced read and write operations on GPU over the unstructured mesh.

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Section: Ebe Strategymentioning

confidence: 99%

Section: Ebe Strategymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…, 2019) and solution of a system of linear equations (Kiran et al. , 2019, 2020). Both direct and iterative solvers have been used for solving the linear system of equations on GPU.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Acceleration of structural topology optimization using symmetric element-by-element strategy for unstructured meshes on GPU

Ratnakar

Kiran²,

Sharma³

2022

Self Cite

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“…Currently, SparseMatrixAssembler, which implements an assembler that builds global sparse matrices from elemental contributions, is the only specialization, but others can be added in the future. E.g., one could easily implement a new Assembler type that builds matrix-free operators [21] instead of sparse matrices.…”

Section: 2mentioning

confidence: 99%

The software design of Gridap: a Finite Element package based on the Julia JIT compiler

Verdugo,

Badia

2021

Preprint

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We present the software design of Gridap, a novel finite element library written exclusively in the Julia programming language, which is being used by several research groups world-wide to simulate complex physical phenomena such as magnetohydrodynamics, photonics, weather modeling, non-linear solid mechanics, and fluid-structure interaction problems. The library provides a feature-rich set of discretization techniques for the numerical approximation of a wide range of mathematical models governed by Partial Differential Equations (PDEs), including linear, nonlinear, single-field, and multi-field equations. An expressive API allows users to define PDEs in weak form by a syntax close to the mathematical notation. While this is also available in previous frameworks, the main novelty of Gridap is that it implements this API without introducing a domain-specific language plus a compiler of variational forms. Instead, it leverages the Julia just-in-time compiler to build efficient code, specialized for the concrete problem at hand. As a result, there is no need to use different languages for the computational back-end and the user front-end anymore, thus eliminating the so-called two-language problem. Gridap also provides a low-level API that is modular and extensible via the multiple-dispatch paradigm of Julia and provides easy access to the main building blocks of the library if required. The main contribution of this paper is the detailed presentation of the novel software abstractions behind the Gridap design that leverages the new software possibilities provided by the Julia language. The second main contribution of the article is a performance comparison against FEniCS. We measure CPU times needed to assemble discrete systems of linear equations for different problem types and show that the performance of Gridap is comparable to FEniCS, demonstrating that the new software design does not compromise performance. Gridap is freely available at Github (github.com/gridap/Gridap.jl) and distributed under an MIT license.

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Development of GPU‐based matrix‐free strategies for large‐scale elastoplasticity analysis using conjugate gradient solver

Kiran,

Sharma,

Gautam

2024

Numerical Meth Engineering

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In recent years, matrix‐free conjugate gradient (CG) solvers with graphics processing unit (GPU) acceleration have been effectively used to reduce the execution timings of finite element method (FEM)‐based engineering simulations. However, there is not much in the literature that discusses the application of matrix‐free CG solvers for elastoplasticity. The primary challenge is the presence of both elastic and plastic states, which introduce branching issues in the parallel computation of sparse matrix‐vector multiplication (SpMV). The current work proposes an efficient split kernel strategy for elastoplasticity that segregates elements in elastic and plastic zones and allows the application of standard GPU‐based matrix‐free SpMV strategies in the literature to elastoplastic simulation. The proposed strategy avoids branching issues, facilitates coalesced memory access, allows efficient usage of on‐chip memory, and uses minimum storage to realize large‐scale simulation on a GPU with limited memory. We also propose matrix‐free SpMV strategies for GPU implementation based on an element‐by‐element technique that makes efficient use of memory hierarchy available in a GPU. The computational efficiency of the proposed strategies is demonstrated over three large‐scale benchmark examples from elastoplasticity, and the performance results are compared with GPU optimized node‐based and degrees of freedom (DOF)‐based matrix‐free strategies. The results show that the splitting of computation is most suitable for low plasticity problems, where most of the matrix‐free strategies show significant speedups with respect to single kernel strategy for elastoplasticity. The proposed element‐by‐element strategy using node‐wise thread allocation is found to have the best performance, achieving up to 7.16 and 3.35 speedup over node‐based and DOF‐based strategy, respectively. For problems with higher amounts of plasticity, the proposed strategies perform slightly better and achieve modest speedups due to advantages in the initial stages of Newton iterations. In terms of wall‐clock timings, the proposed matrix‐free solver is found to be up to 2 faster than GPU optimized assembly‐based elastoplasticity solver.

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GPU-based matrix-free finite element solver exploiting symmetry of elemental matrices

Cited by 23 publications

References 38 publications

Acceleration of structural topology optimization using symmetric element-by-element strategy for unstructured meshes on GPU

Acceleration of structural topology optimization using symmetric element-by-element strategy for unstructured meshes on GPU

The software design of Gridap: a Finite Element package based on the Julia JIT compiler

Development of GPU‐based matrix‐free strategies for large‐scale elastoplasticity analysis using conjugate gradient solver

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