Physics-based balancing domain decomposition by constraints for multi-material problems

The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-018-0870-zIn this work, we present a new variant of the balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced subdomain partition, and based on an aggregation of elements according to their physical coefficients, we end up with a finer physics-based (PB) subdomain partition. Next, we define corners, edges, and faces for this PB partition, and select some of them to enforce subdomain continuity (primal faces/edges/corners). When the physical coefficient in each PB subdomain is constant and the set of selected primal faces/edges/corners satisfy a mild condition on the existence of acceptable paths, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coefficient across subdomains. An extensive set of numerical experiments for 2D and 3D for the Poisson and linear elasticity problems is provided to support our findings. In particular, we show robustness and weak scalability of the new preconditioner variant up to 8232 cores when applied to 3D multi-material problems with the contrast of the physical coefficient up to 108 and more than half a billion degrees of freedom. For the scalability analysis, we have exploited a highly scalable advanced inter-level overlapped implementation of the preconditioner that deals very efficiently with the coarse problem computation. The proposed preconditioner is compared against a state-of-the-art implementation of an adaptive BDDC method in PETSc for thermal and mechanical multi-material problems.Peer ReviewedPostprint (author's final draft

Physics-based balancing domain decomposition by constraints for multi-material problems

In this work, we present a new variant of the balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced subdomain partition, and based on an aggregation of elements according to their physical coefficients, we end up with a finer physics-based (PB) subdomain partition. Next, we define corners, edges, and faces for this PB partition, and select some of them to enforce subdomain continuity (primal faces/edges/corners). When the physical coefficient in each PB subdomain is constant and the set of selected primal faces/edges/corners satisfy a mild condition on the existence of acceptable paths, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coefficient across subdomains. An extensive set of numerical experiments for 2D and 3D for the Poisson and linear elasticity problems is provided to support our findings. In particular, we show robustness and weak scalability of the new preconditioner variant up to 8232 cores when applied to 3D multi-material problems with the contrast of the physical coefficient up to 108 and more than half a billion degrees of freedom. For the scalability analysis, we have exploited a highly scalable advanced inter-level overlapped implementation of the preconditioner that deals very efficiently with the coarse problem computation. The proposed preconditioner is compared against a state-of-the-art implementation of an adaptive BDDC method in PETSc for thermal and mechanical multi-material problems

Balancing domain decomposition by constraints associated with subobjects

A simple variant of the BDDC preconditioner in which constraints are imposed on a selected set of subobjects (subdomain subedges, subfaces and vertices between pairs of subedges) is presented. We are able to show that the condition number of the preconditioner is bounded by C 1 + log(L/h)2, where C is a constant, and h and L are the characteristic sizes of the mesh and the subobjects, respectively. As L can be chosen almost freely, the condition number can theoretically be as small as O(1). We will discuss the pros and cons of the preconditioner and its application to heterogeneous problems. Numerical results on supercomputers are provided

Balancing domain decomposition by constraints associated with subobjects

A simple variant of the BDDC preconditioner in which constraints are imposed on a selected set of subobjects (subdomain subedges, subfaces and vertices between pairs of subedges) is presented. We are able to show that the condition number of the preconditioner is bounded by C(1+log(L/h))2, where C is a constant, and h and L are the characteristic sizes of the mesh and the subobjects, respectively. As L can be chosen almost freely, the condition number can theoretically be as small as O(1). We will discuss the pros and cons of the preconditioner and its application to heterogeneous problems. Numerical results on supercomputers are provided.Peer ReviewedPostprint (author's final draft