Toward GPU accelerated topology optimization on unstructured meshes

Abstract

The present work investigates the feasibility of finite element methods and topology optimization for unstructured meshes in massively parallel computer architectures, more specifically on Graphics Processing Units or GPUs. Challenges in the parallel implementation, like the parallel assembly race condition, are discussed and solved with simple algorithms, in this case greedy graph coloring. The parallel implementation for every step involved in the topology optimization process is benchmarked and compared against an equivalent sequential implementation. The ultimate goal of this work is to speed up the topology optimization process by means of parallel computing using off-the-shelf hardware. Examples are compared with both a standard sequential version of the implementation and a massively parallel version to better illustrate the advantages and disadvantages of this approach.

Appendix A: Nomenclature

1.1 A.1 Symbols

${[\mathbf {B}]}$ :

Strain-displacement matrix

$b_w$ :

Bandwidth of a matrix

c:

Compliance

${[\mathbf {D}]}$ :

Constitutive matrix

E:

Young’s modulus

f:

Volume fraction

${\{\mathbf {f}\}}$ :

Global force vector

$\hat {H}_j$ :

Convolution function

${[\mathbf {J}]}$ :

Transformation Jacobian

${[\mathbf {K}]}$ :

Global stiffness matrix

${[\mathbf {k}]}$ :

Local stiffness matrix

${[\mathbf {L}]}$ :

Lower triangular matrix

${\mathbf {L} ( G^C )}$ :

Communication matrix for graph G

m:

Density move limit

n:

Number of elements, matrix size or other depending on the context

p:

Penalization factor for SIMP

R:

Filter radius

u:

Displacement

V:

Volume

${[\mathbf {XY}]}$ :

Nodal coordinates of an element

${[\boldsymbol {\Gamma }]}$ :

Inverse of the transformation Jacobian$[\mathbf {J}]$

$\eta$ :

Numerical damping parameter

$\mathcal {L}$ :

Lagrangian function

$\lambda$ :

Lagrange multiplier

$\nu$ :

Poisson’s ratio

$\rho$ :

Density

${\chi ( G )}$ :

Chromatic number for graph G

1.2 A.2 Abbreviations

ALU:

Arithmetic Logic Unit

CAMD:

Continuous Approximation of Material Distribution

CPU:

Central Processing Unit

CUDA:

NVIDIA’s Compute Unified Device Architecture

DOF:

Degree Of Freedom

DRAM:

Dynamic Random-Access Memory

FEM:

Finite Element Method

GPU:

Graphics Processing Unit

MBB:

Messerschmitt–Bölkow–Blohm

OC:

Optimality Criteria

PCGS:

Pre-conditioned Conjugate Gradient Solver

SIMP:

Solid Isotropic Material with Penalization

TOP:

Topology Optimization