A Cross-Cutting Excalibur Project
A Cross-Cutting Excalibur Project
Topics include (but are not restricted to) multi-physics coupling problems, interface problems, multi-domain problems, preconditioning of high frequency scattering, multi-methods coupling (such as FEM/BEM).
All times in GMT
Registration is required to access the Zoom Webinar
In this talk I will present a well-conditioned weak coupling of boundary element and high-order finite element methods for time-harmonic electromagnetic scattering by inhomogeneous objects [1]. The approach is based on the use of a non-overlapping domain decomposition method involving quasi-optimal transmission operators. The associated transmission boundary conditions are constructed through a localization process based on complex rational Padé approximants of the nonlocal Magnetic-to-Electric operators. The number of iterations required to solve this weak coupling is only slightly dependent on the frequency and the mesh refinement.
[1] Badia, I., Caudron, B., Antoine, X., & Geuzaine, C. (2022). A well-conditioned weak coupling of boundary element and high-order finite element methods for time-harmonic electromagnetic scattering by inhomogeneous objects. SIAM Journal on Scientific Computing, 44(3), B640-B667.
We introduce a scalable adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom along with comparisons to Algebraic Multigrid Methods.
We present the basic concepts in CutFEM, including stabilization methods and recent developments on so called extension operators which essentially eliminates unstable degrees of freedom in such a way that optimal approximation properties are retained. Then we turn to hybridization and derive a domain decomposition method and discuss various ways of choosing the meshes. Finally, we extend the hybridization approach to model coupling in computational mechanics.
Mathematical problems describing the mechanical interaction of a flexible structure with an incompressible fluid flow appear in a wide variety of engineering fields. Fluid-structure interaction is also particularly ubiquitous in nature. One can think, for instance, of the wings of a bird interacting with the air, the fins of a fish moving through the water, or blood propelled into the arteries. The solid is deformed under the action of the fluid and the fluid flow is disturbed by the moving solid. Such multi-physic phenomena are generally described by heterogeneous systems of non-linear equations with an interface coupling which can be extremely stiff when solved via partitioned solution procedures (the so-called added-mass effect). Over the last fifteen years, the development and analysis of efficient partitioned methods for these systems has been a very active field of research. In this talk, we will give an overview of some of these techniques, with particular emphasis on time splitting schemes.
We discuss, in a general (abstract) framework, and with concrete examples, the numerical solution of a global problem formulated, already at the continuous level, as a system of weakly coupled local problems. With an approach similar to the one underlying the FETI domain decomposition method and treating the local numerical solvers as black-boxes, we discuss sufficient conditions ensuring stability and accuracy of the discrete coupled problem, for which we also discuss a preconditioning strategy (also inspired by FETI).
For the eddy current approximation of the Maxwell system to model an electric motor results in an elliptic-parabolic interface problem, we introduce and analyze a space-time variational formulation, and its space-time finite element discretization. In addition we also discuss the coupling with boundary elements to model the Laplace equation in the air domain. For this we recall some results on the non-symmetric coupling of finite and boundary element methods.