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FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D
(2022)
We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.
Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction
(2019)
The paper presents a fully-implicit and stable finite element and finite volume scheme for the simulation of freely moving particles in a fluid. The developed method is based on the Petrov-Galerkin formulation of a vertex-centered finite volume method (PG-FVM) on unstructured grids. Appropriate extension of the ansatz and test spaces lead to a formulation comparable to a fictitious domain formulation. The purpose of this work is to introduce a new concept of numerical modeling reducing the mathematical overhead which many other methods require. It exploits the identification of the PG-FVM with a corresponding finite element bilinear form. The surface integrals of the finite volume scheme enable a natural incorporation of the interface forces purely based on the original bilinear operator for the fluid. As a result, there is no need to expand the system of equations to a saddle-point problem. Like for fictitious domain methods the extended scheme treats the particles as rigid parts of the fluid. The distinguishing feature compared to most existing fictitious domain methods is that there is no need for an additional Lagrange multiplier or other artificial external forces for the fluid-solid coupling. Consequently, only one single solve for the derived linear system for the fluid together with the particles is necessary and the proposed method does not require any fractional time stepping scheme to balance the interaction forces between fluid and particles. For the linear Stokes problem we will prove the stability of both schemes. Moreover, for the stationary case the conservation of mass and momentum is not violated by the extended scheme, i.e. conservativity is accomplished within the range of the underlying, unconstrained discretization scheme. The scheme is applicable for problems in two and three dimensions.