312 research outputs found
The conforming virtual element method for polyharmonic and elastodynamics problems: a review
In this paper, we review recent results on the conforming virtual element
approximation of polyharmonic and elastodynamics problems. The structure and
the content of this review is motivated by three paradigmatic examples of
applications: classical and anisotropic Cahn-Hilliard equation and phase field
models for brittle fracture, that are briefly discussed in the first part of
the paper. We present and discuss the mathematical details of the conforming
virtual element approximation of linear polyharmonic problems, the classical
Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1912.0712
Virtual Element Methods for hyperbolic problems on polygonal meshes
In the present paper we develop the Virtual Element Method for hyperbolic
problems on polygonal meshes, considering the linear wave equations as our
model problem. After presenting the semi-discrete scheme, we derive the
convergence estimates in H^1 semi-norm and L^2 norm. Moreover we develop a
theoretical analysis on the stability for the fully discrete problem by
comparing the Newmark method and the Bathe method. Finally we show the
practical behaviour of the proposed method through a large array of numerical
tests
A decoupled, stable, and linear FEM for a phase-field model of variable density two-phase incompressible surface flow
The paper considers a thermodynamically consistent phase-field model of a
two-phase flow of incompressible viscous fluids. The model allows for a
non-linear dependence of fluid density on the phase-field order parameter.
Driven by applications in biomembrane studies, the model is written for
tangential flows of fluids constrained to a surface and consists of (surface)
Navier-Stokes-Cahn-Hilliard type equations. We apply an unfitted finite element
method to discretize the system and introduce a fully discrete time-stepping
scheme with the following properties: (i) the scheme decouples the fluid and
phase-field equation solvers at each time step, (ii) the resulting two
algebraic systems are linear, and (iii) the numerical solution satisfies the
same stability bound as the solution of the original system under some
restrictions on the discretization parameters. Numerical examples are provided
to demonstrate the stability, accuracy, and overall efficiency of the approach.
Our computational study of several two-phase surface flows reveals some
interesting dependencies of flow statistics on the geometry.Comment: 22 pages, 5 figures, 1 tabl
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
It is well known that the solution of topology optimization problems may be
affected both by the geometric properties of the computational mesh, which can
steer the minimization process towards local (and non-physical) minima, and by
the accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
Computational phase-field modeling
Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community
Virtual element methods for fourth-order problems : implementation and analysis
In this thesis we aim to create a unified framework for the implementation and analysis of virtual element spaces. The approach we take for the virtual element discretisation allows us to easily construct vector field spaces as well as consider both variable coefficient and nonlinear problems. On top of this, the approach can be integrated more readily into existing finite element software packages. These are significant advantages of the method we present and something that has not been easy to achieve within the original virtual element setting. We exploit these key advantages in this thesis. In particular, we present a straightforward and generic way to define the projection operators, which form a crucial part of the virtual element discretisation, for a wide range of problems. We demonstrate how to build Hm-conforming for m = 1, 2 and nonconforming spaces as well as divergence and curl free spaces. All of which have been implemented in the open source Dune software framework as part of the Dune-Fem module. As a consequence of the projection approach taken in our framework, we are able to carry out a priori error analysis for higher order approximations of the following fourth-order problems: a general linear fourth-order PDE with non-constant coefficients; a singular perturbation problem; and the nonlinear time-dependent Cahn-Hilliard equation. Furthermore, we showcase the versatility of the projection approach with the introduction of a novel nonconforming scheme for the singular perturbation problem. The modified nonconforming method is uniformly convergent with respect to the perturbation parameter and unlike modifications in the literature, does not require an enlargement of the space. Numerical tests are carried out to verify the theoretical results
Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects
In this paper, we consider a coupled PDE system describing phase separation
and damage phenomena in elastically stressed alloys in the presence of inertial
effects. The material is considered on a bounded Lipschitz domain with mixed
boundary conditions for the displacement variable. The main aim of this work is
to establish existence of weak solutions for the introduced
hyperbolic-parabolic system. To this end, we first adopt the notion of weak
solutions introduced in [C. Heinemann, C. Kraus: Existence results of weak
solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv.
Math. Sci. Appl. 21 (2011), 321--359]. Then we prove existence of weak
solutions by means of regularization, time-discretization and different
variational techniques
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