arXiv subjects

G. N. Wells

Publications and source records attributed to G. N. Wells.

3 recordsLinked to original sources

arXiv2024

Mixed-precision finite element kernels and assembly: Rounding error analysis and hardware acceleration

In this paper we develop the first fine-grained rounding error analysis of finite element (FE) cell kernels and assembly. The theory includes mixed-precision implementations and accounts for hardware-acceleration via matrix multiplication units, thus providing theoretical guidance for designing reduced- and mixed-precision FE algorithms on CPUs and GPUs. Guided by this analysis, we introduce hardware-accelerated mixed-precision implementation strategies which are provably robust to low-precision computations. Indeed, these algorithms are accurate to the lower-precision unit roundoff with an error constant that is independent from: the conditioning of FE basis function evaluations, the ill-posedness of the cell, the polynomial degree, and the number of quadrature nodes. Consequently, we present the first AMX-accelerated FE kernel implementations on Intel Sapphire Rapids CPUs. Numerical experiments demonstrate that the proposed mixed- (single/half-) precision algorithms are up to 60 times faster than their double precision equivalent while being orders of magnitude more accurate than their fully half-precision counterparts.

math.NA↗

arXiv2004

A discontinuous Galerkin method for strain gradient-dependent damage: Study of interpolations, convergence and two dimensional problems

A discontinuous Galerkin method has been developed for strain gradient-dependent damage. The strength of this method lies in the fact that it allows the use of $C^0$ interpolation functions for continuum theories involving higher-order derivatives, while in a conventional framework at least $C^1$ interpolations are required. The discontinuous Galerkin formulation thereby offers significant potential for engineering computations with strain gradient-dependent models. When using basis functions with a low degree of continuity, jump conditions arise at element edges which are incorporated in the weak form. In addition to the formulation itself, a detailed study of the convergence properties of the method for various element types is presented, an error analysis is undertaken, and the method is also shown to work in two dimensions.

physics.comp-ph↗

arXiv2003

A discontinuous Galerkin formulation for a strain gradient-dependent damage model

The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand $C^{1}$ continuity of the shape functions. Here, recent development in discontinuous Galerkin methods are explored and exploited for the solution of a prototype nonlinear strain gradient dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard $C^{0}$ shape functions. The formulation is tested in one-dimension for the simplest possible finite element formulation: piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed formulation.

math.NA↗