Local nonreflecting boundary condition for time-dependent multiple scattering

Starting from from a high-order local nonreflecting boundary condition (NBC) for single scattering [25], we derive a local NBC for time-dependent multiple scattering problems in three space dimensions, which is completely local both in space and time. To do so, we first develop an exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NBC at the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain a completely local NBC for time-dependent multiple scattering problems. The accuracy and stability of this new local NBC are evaluated by coupling it to a standard finite difference method. (C) 2011 Elsevier Inc. All rights reserved.

Adaptive finite elements for a linear parabolic problem

Marco Picasso

A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach. (C) 1998 Elsevier Science S.A. All rights reserved.

1998

Fluid-structure interaction for vascular flows: From supercomputers to laptops

Claudia Maria Colciago, Simone Deparis, Davide Forti

Several models exist for the simulation of vascular flows; they span from simple circuit models to full three-dimensional ones that take into account detailed features of the blood and of the arterialwall. Eachmodel comeswith both benefits and drawbacks, the main denominator being a compromise between detailed resolution requirements versus computational time. We first present a fluid-structure interaction computationalmodelwhere both the fluid and the structure are three dimensional. In particular, the fluid includes modeling of large eddies by the variationalmultiscalemethod. After time and space discretizations carried out by finite differences and finite elements, respectively, we set up a parallel solver based ondomain decomposition and a FaCSI preconditioner. These simulations allow one to capture details of the flow dynamics and of the structure deformation even in the transitional regime characterizing hemodynamics in the aorta. It takes roughly 10 hours to complete a simulation of one heartbeat with 35 million degrees of freedom on 2048 cores. We then reduce both the model and its numerical complexity. The structural model is simplified to a two-dimensional membrane located at the fluid-structure interface and the fluid computational domain is fixed. For a fixed geometry andmesh, these assumptions allow one to apply proper orthogonal decomposition and generate a space discretization which has only a few dozen degrees of freedom. It is then possible to perform the simulation of one heartbeat on a laptop in less than one second. Themodeling and numerical reduction therefore allows a dramatic reduction of computational time. However, the price to pay comes, on the one hand, in terms of the preparation of a reduced basis specific to the patient and the geometry of the vessel and, on the other hand, with a detriment of certain quantities of interest. For example, when using a finite element discretization with 9 million degrees of freedom, the offline part takes about 12 hours on 720 cores for the example provided in this work; in this case, the flow profiles in the aorta are pretty close to the full three-dimensional

De Gruyter2017

MATHICSE Technical Report : Fast solvers for 2D fractional differential equations using rank structured matrices

Stefano Massei, Marco Mazza

We consider the discretization of time-space diffusion equations with fractional derivatives in space and either 1D or 2D spatial domains. The use of implicit Euler scheme in time and finite differences or finite elements in space, leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats (H-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitz-like structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the

MATHICSE2018

Fluid-structure interaction for vascular flows: From supercomputers to laptops

Claudia Maria Colciago, Simone Deparis, Davide Forti

De Gruyter2017