A Parametric Finite Element Method for a Degenerate Multi-Phase Stefan Problem with Triple Junctions

This document presents a description of the octree mesh-generation capabilities and of the parallel mesh adaptation kernel. As it is discussed in Section 1.3.2 of part B of the project proposal there are two parallel research lines aimed at developing scalable adaptive mesh refinement (AMR) algorithms and implementations. The first one is based on using octree-based mesh generation and adaptation for the whole simulation in combination with unfitted finite element methods (FEMs) and the use of algebraic constraints to deal with non-conformity of spaces. On the other hand the second strategy is based on the use of an initial octree mesh that, after make it conforming through the addition of templatebased tetrahedral refinements, is adapted anisotropically during the calculation. Regarding the first strategy the following items are included:

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. The method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping are easy to implement. In this paper we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high order discretization. The theoretical results are confirmed in numerical experiments.

A generalized multigrid method for solving contact problems in Lagrange multiplier based unfitted Finite Element method

In this paper we propose a method for the finite element solution of elliptic interface problem, using an approach due to Nitsche. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. Further, we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples.

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

Analysis of higher-order corrections to the interface kinetics in the multiphase field method

In this work we examine the existence of positive classical solutions of { − Δ u + u = | x | α u p − 1 a m p ; in B 1 , u > 0 a m p ; in B 1 , ∂ ν u = 0 a m p ; on ∂ B 1 , \begin{equation*} \begin {cases} -\Delta u +u = |x|^\alpha u^{p-1} & \text { in } B_1, \\ u>0 & \text { in } B_1, \\ \partial _\nu u= 0 & \text { on } \partial B_1, \end{cases} \end{equation*} where p > 1 p>1 , α > 0 \alpha >0 and B 1 B_1 is the unit ball in R N {\mathbb {R}}^N where N ≥ 4 N \ge 4 and is even. Of particular interest is the existence of nonradial position classical solutions. We show that under suitable conditions on p , α p,\alpha and N N there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones.

We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain threshold, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of ${C}^1$ initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives should be large. Second, when the initial data have constant potential vorticity, global existence of small classical solutions is established via studying an equivalent form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global existence result, the smallness condition is in terms of the higher order Sobolev norms of the initial data.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Numerical study of the grain-size dependent Young’s modulus and Poisson’s ratio of bulk nanocrystalline materials

Some optimal design problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):=\int_{\Omega}W(\nabla v)dx-\int_{\Omega}f\,v\,dx$ for $v \in H_0^1(\Omega)$ with possibly multiple minimizers $u$, but with a unique stress $\sigma:=DW(\nabla u)$. This paper proposes the discrete Raviart--Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix--Raviart nonconforming finite element method. The convergence analysis combines the a priori convergence rate of the conforming FEM with the efficient a posterior error control of MFEM. Numerical experiments provide empirical evidence that the proposed dRT-MFEM overcomes the reliability-efficiency gap for the first time.

We prove the existence of exact solutions and convergence of numerical (Galerkin) approximations for the semilinear degenerate parabolic problem ut+Lu = F(t,u) and for the degenerate elliptic problem Lu = f where L is a highly singular linear elliptic operator. The class of operators L contains in particular the operators of generalized bi-axially symmetric potential theory.

Based on the classical plate theory in conjunction with the assumption of line contact at the compressive edge of a crack face, closed form solutions were presented for a through-the-thickness central crack in an infinite plate subjected to all around bending. The complete solutions were obtained by superposing the membrane components due to the contact forces at the crack face to the non-closure bending components. The distribution of the contact forces was found uniform by considering the contact condition which prevents mutual penetration of the crack faces at the compressive edges. The results showed that the closure of the crack faces tends to reduce the crack opening displacement at the tension side and, consequently, reduce the stress intensity factor. The finite element method was also used to investigate the present problem. The modified crack closure method in combination with the finite element method was used to find the stress intensity factors. Close agreement between the finite element and the analytical solutions was observed.

A coupling approach of discretized peridynamics with finite element method