The Deflated Preconditioned Conjugate Gradient Method Applied to Composite Materials
Tom B. Jönsthövel · Research Repository (Delft University of Technology) · 2012
Simulations with composite materials often involve large jumps in the coefficients of the underlying stiffness matrix. These jumps can introduce unfavorable eigenvalues in the spectrum of the stiffness matrix. We show that the rigid body modes; the translations and rotations, of the disjunct rigid bodies in the composite material correspond to these unfavorable eigenvalues. The stiffness matrix is symmetric positive definite and therefore the preconditioned conjugate gradient (PCG) method is the method of choice for solving the linear systems involved. The unfavorable eigenvalues that correspond to the rigid body modes, slow down the convergence of PCG. By using rigid body mode deflation, the unfavorable eigenvalues are removed from the spectrum of the stiffness matrix. We construct a robust and fast iterative solver, the deflated preconditioned conjugate gradient (DPCG) method, to efficiently solve the linear systems involved. The DPCG method is well suited for parallel computations. Moreover, we show that the DPCG method is a very competitive method compared to other state-of-the-art (parallel) linear solvers such as smoothed aggregation algebraic multigrid and sparse direct solvers.