Large Scale Finite Element Analysis Via Assembly-Free Deflated Conjugate Gradient

Abstract

Large-scale finite element analysis (FEA) with millions of degrees of freedom (DOF) is becoming commonplace in solid mechanics. The primary computational bottleneck in such problems is the solution of large linear systems of equations. In this paper, we propose an assembly-free version of the deflated conjugate gradient (DCG) for solving such equations, where neither the stiffness matrix nor the deflation matrix is assembled. While assembly-free FEA is a well-known concept, the novelty pursued in this paper is the use of assembly-free deflation. The resulting implementation is particularly well suited for large-scale problems and can be easily ported to multicore central processing unit (CPU) and graphics-programmable unit (GPU) architectures. For demonstration, we show that one can solve a 50 × 106 degree of freedom system on a single GPU card, equipped with 3 GB of memory. The second contribution is an extension of the “rigid-body agglomeration” concept used in DCG to a “curvature-sensitive agglomeration.” The latter exploits classic plate and beam theories for efficient deflation of highly ill-conditioned problems arising from thin structures.

A Pareto-Optimal Approach to Multimaterial Topology Optimization

“…Further, we consider a simple finite element discretization, where the geometry is approximated via uniform hexahedral elements or 'voxels'; the voxel-approach has gained significant popularity recently due to its robustness and low memory foot-print. The more significant benefits of voxelization are: (1) it is robust in that the automatic mesh generation rarely fails (unlike traditional meshing), (2) the mesh storage is compact, (3) the cost of voxelization is usually negligible and is relatively insensitive to geometric complexity, (4) it promotes assemblyfree-FEA [62], [63], and (5) it simplifies the proposed optimization algorithm. Moreover, we use conjugate gradient (CG) method to solve the FEA linear system of equations.…”

“…Deflation is a powerful acceleration technique for CG, and is more suitable for the assembly-free FEA than classic preconditioners such as incomplete Cholesky. In this paper, we use a deflation method based on rigid-body deflation presented in [62].…”

“…In all tables, E , ν , and ρ denote Young's modulus, Poisson ratio, and density, respectively. For all experiments, we rely on the assembly-free deflated conjugate gradient (CG) discussed in [62], with a tolerance set to 10 -8 .…”

Topsim: A Plugin-Based Framework for Large-Scale Numerical Analysis

“…They presented results for the compliance minimization problem with more than 100 million elements, using 1800 cores. In 2014, Yadav and Suresh [27] presented an assembly-free version of the deflated conjugate gradient (DCG) for solving large linear system of equations, where neither the stiffness matrix nor the deflation matrix is assembled. The novelty pursued in the paper is the use of assembly-free deflation.…”

“…Foi simulado um exemplo de análise estática linear elástica com 500 milhões de elementos em 300 máquinas. 27 Time to solve the 3D Cantilever Beam problem using the iterative solver from Abaqus and the PCG from TopSim, both considering only 1 core of the machine. 57 3.28 Time to solve the 3D Cantilever Beam problem using the direct solver of Abaqus and the PARDISO solver of TopSim, both using only 1 core of the machine.…”