Parallel Fast Isogeometric Solvers for Explicit Dynamics

This paper presents a parallel implementation of the fast isogeometric solvers for explicit dynamics for solving non-stationary time-dependent problems. The algorithm is described in pseudo-code. We present theoretical estimates of the computational and communication complexities for a single time step of the parallel algorithm. The computational complexity is O(p^6 N/c t_comp) and communication complexity is O(N/(c^(2/3)t_comm) where p denotes the polynomial order of B-spline basis with Cp-1 global continuity, N denotes the number of elements and c is number of processors forming a cube, t_comp refers to the execution time of a single operation, and t_comm refers to the time of sending a single datum. We compare theoretical estimates with numerical experiments performed on the LONESTAR Linux cluster from Texas Advanced Computing Center, using 1 000 processors. We apply the method to solve nonlinear flows in highly heterogeneous porous media

Phase-field modeling of planar interface electrodeposition in lithium-metal batteries

Acknowledgments This work was supported by the Aberdeen-Curtin Alliance Scholarship. This publication was also made possible in part by the Professorial Chair in Computational Geoscience at Curtin University. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS). The Curtin Corrosion Centre and the Curtin Institute for Computation kindly provide ongoing support.Peer reviewedPostprin

Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines

This paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the Cp-1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver are of order O(log(N)p2) for the one dimensional (1D) case, O(Np2) for the two dimensional (2D) case, and O(N4/3p2) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N. For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20% for 256 processors for a 3D example with 1283 unknowns and linear B-splines with C0 global continuity, and 15% for a 3D example with 643 unknowns and quartic B-splines with C3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

We provide explicit quadrature rules for spaces of C1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. Each rule is optimal, that is, requires the minimal number of nodes, for a given function space. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2/3 when compared to the classical Gaussian quadrature for polynomials over each knot span. Numerical experiments show fast convergence, as n grows, to the “two-third” quadrature rule of Hughes et al. (2010) for infinite domains

Self-assembly in casting solutions of block copolymer membranes

Membranes with exceptional pore regularity and high porosity were obtained from block copolymer solutions. We demonstrate by small-angle X-ray scattering that the order which gives rise to the pore morphology is already incipient in the casting solution. Hexagonal order was confirmed in PS-b-P4VP 175k-b-65k solutions in DMF/THF/dioxane with concentrations as high as 24 wt%, while lamellar structures were obtained in more concentrated solutions in DMF or DMF/dioxane. The change in order has been understood with the support of dissipative particle dynamic modeling