David Wiedemann

TU Dortmund University | TUD · Department of Mathematics

We give a proof for the existence of weak solutions to time-harmonic Maxwell's equations on bounded Lipschitz domains. This result is well-known; our aim here is to present, in a concise way, the arguments that lead from a compactness result for divergence free functions and a Helmholtz decomposition to the existence result.

We consider the homogenisation of a diffusion equation in a porous medium. The microstructure is time-dependent and oscillating on a small time scale. This oscillation causes a novel advection in the homogenised equations. Allowing for a locally varying geometry, the oscillating microstructure demonstrates the ability to generate arbitrary and loca...

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope...

We present a singular value polyconvex conjugation. Employing this conjugation, we derive a necessary and sufficient criterion for polyconvexity of isotropic functions by means of the convexity of a function with respect to the signed singular values. Moreover, we present a new criterion for polyconvexity of isotropic functions by means of matrix i...

We consider the homogenisation of a coupled reaction–diffusion process in a porous medium with evolving microstructure. A concentration-dependent reaction rate at the interface of the pores with the solid matrix induces a concentration-dependent evolution of the domain. Hence, the evolution is fully coupled with the reaction–diffusion process. In o...

We consider the homogenisation of a coupled reaction-diffusion process in a porous medium with evolving microstructure. A concentration-dependent reaction rate at the interface of the pores with the solid matrix induces a concentration-dependent evolution of the domain. Hence, the evolution is fully coupled with the reaction-diffusion process. In o...

We present the two-scale-transformation method which allows rigorous homogenisation of problems defined in locally periodic domains. This method transforms such problems into periodic domains in order to facilitate the passage to the limit. The idea of transforming problems into periodic domains originates from the homogenisation of problems define...

A Darcy law for evolving microstructure is derived by homogenisation of Stokes flow. We transform the Stokes equations from the locally evolving domain onto a periodic reference domain. There, we pass to the homogenisation limit by employing the method of two-scale convergence. After transforming the limit back to its original two-scale domain, we...

We consider the homogenisation of the Stokes equations in a porous medium which is evolving in time. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables to model a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit with the...

We prove the two-scale transformation method which allows rigorous homogenisation of problems defined on locally periodic domains by transformation on periodic domains. The idea to consider periodic substitute problems was originally proposed by M. A. Peter for the homogenisation on evolving microstructure and is applied in several works. However,...