Related papers: Sharp interface limit for a phase field model in s…
Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which…
The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity…
A phase-field approach describing the dynamics of a strained solid in contact with its melt is developed. By rigorous asymptotic analysis we show that the sharp-interface limit of this model recovers the continuum model equations for the…
We consider a continuum mechanical model of cell invasion through thin membranes. The model consists of a transmission problem for cell volume fraction complemented with continuity of stresses and mass flux across the surfaces of the…
In this paper we study a singular control problem for a system of PDEs describing a phase-field model of Penrose-Fife type. The main novelty of this contribution consists in the idea of forcing a sharp interface separation between the…
In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape…
We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. Our aim is a scalable algorithm for application on supercomputers, which can only be achieved by mesh-independent convergence. The impact of…
In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional…
Topology optimization has matured to become a powerful engineering design tool that is capable of designing extraordinary structures and materials taking into account various physical phenomena. Despite the method's great advancements in…
In the present article we study diffuse interface models for two-phase biomembranes. We will do so by starting off with a diffuse interface model on $\mathbb{R}^n$ defined by two coupled phase fields $u,v$. The first phase field $u$ is the…
The phase-field method is reviewed from the general perspective of converting a free boundary problem into a set of coupled partial differential equations. Its main advantage is that it avoids front tracking by using phase fields to locate…
We study finite element approximations of elliptic and parabolic interface problems with discontinuous coefficients and nonlinear jump conditions. We introduce a scalar interface reduction in which the solution is decomposed into a…
In this paper, we study the sharp interface limit for solutions of the Cahn-Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing…
We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic…
The modeling of crack propagation in a heterogeneous material using a phase field model is studied numerically in a simple test case: the crack meets a wedge of higher fracture energy. It is shown that when the crack cannot enter the wedge,…
In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form: $$\partial_t A=\Delta A-\varepsilon^{-2}( A A^{\mathrm{T}}A-…
In this work we consider topology optimization of systems, which are governed by the external Bernoulli free boundary problem. We utilize the so-called pseudo-solid approach to solve the governing free boundary problems during the…
We describe the implementation of a topological constraint in finite element simulations of phase field models which ensures path-connectedness of preimages of intervals in the phase field variable. Two main applications of our method are…
This paper sets up an approach for shape optimization problems constrained by variational inequalities (VI) in an appropriate shape space. In contrast to classical VI, where no explicit dependence on the domain is given, VI constrained…
In traditional phase-field modeling of multiphase materials, a significant challenge arises from the non-local nature of fracture energy regularization, where interfacial toughness is inherently coupled with the properties of the…