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We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility of discontinuous Galerkin methods on unstructured meshes. The two spatial discretizations are coupled by a penalty technique at the interface such that the overall semidiscretization satisfies a discrete energy estimate to ensure stability. In addition, optimal convergence is obtained in the sense that when combining a fourth order finite difference method with a discontinuous Galerkin method using third order local polynomials, the overall convergence rate is fourth order. Furthermore, we use a novel approach to derive an error estimate for the semidiscretization by combining the energy method and the normal mode analysis for a corresponding one dimensional model problem. The stability and accuracy analysis are verified in numerical experiments.

This paper presents a stable finite element approximation for the acoustic wave equation on second-order form, with perfectly matched layers (PML) at the boundaries. Energy estimates are derived for varying PML damping for both the discrete and the continuous case. Moreover, a priori error estimates are derived for constant PML damping. Most of the analysis is performed in Laplace space. Numerical experiments in physical space validate the theoretical results., 30 pages

It is well-known that reliable and efficient domain truncation is crucial to accurate numerical solution of most wave propagation problems. The perfectly matched layer (PML) is a method which, when stable, can provide a domain truncation scheme which is convergent with increasing layer width/damping. The difficulties in using the PML are primarily associated with stability, which can be present at the continuous level or be triggered by numerical approximations. The mathematical and numerical analysis of the PML for hyperbolic wave propagation problems has been an area of active research. It is now possible to construct stable and high order accurate numerical wave solvers by augmenting wave equations with the PML and approximating the equations using summation-by-parts finite difference methods, continuous and discontinuous Galerkin finite element methods. In this review we summarise the progress made, from mathematical, numerical and practical perspectives, point out some open problems and set the stage for future work. We also present numerical experiments of model problems corroborating the theoretical analysis, and numerical simulations of real-world wave propagation demonstrating impact. Stable and parallel implementations of the PML in the high performance computing software packages WaveQLab3D and ExaHyPE allow to sufficiently limit the computational domain of seismological problems with only a few grid points/elements around the computational boundaries where the PML is active, thus saving as much as $96\%$ of the required computational resources for a three space dimensional seismological benchmark problem.

Atomistic-continuum multiscale modelling is becoming an increasingly popular tool for simulating the behaviour of materials due to its computational efficiency and reliable accuracy. In the case of ferromagnetic materials, the atomistic approach handles the dynamics of spin magnetic moments of individual atoms, while the continuum approximations operate with volume-averaged quantities, such as magnetisation. One of the challenges for multiscale models in relation to physics of ferromagnets is the existence of the long-range dipole-dipole interactions between spins. The aim of the present paper is to demonstrate a way of including these interactions into existing atomistic-continuum coupling methods based on the partitioned-domain and the upscaling strategies. This is achieved by modelling the demagnetising field exclusively at the continuum level and coupling it to both scales. Such an approach relies on the atomistic expression for the magnetisation field converging to the continuum expression when the interatomic spacing approaches zero, which is demonstrated in this paper.

The conventional no-slip boundary condition leads to a non-integrable stress singularity at a contact line. This is a main challenge in numerical simulations of two-phase flows with moving contact lines. We derive a two-dimensional hydrodynamic model for the velocity field at a contact point moving with constant velocity over a curved wall. The model is a perturbation of the classical Huh and Scriven hydrodynamic solution [11], which is only valid for flow over a flat wall. The purpose of the hydrodynamic model is to investigate the macroscopic behavior of the fluids close to a contact point. We also present an idea for how the hydrodynamic solution could be used to prescribe macroscopic Dirichlet boundary conditions for the velocity in the vicinity of a moving contact point. Simulations demonstrate that the velocity field based on the non-singular boundary conditions is capable of accurately advecting the contact point.

A high order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface are allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche's method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts.

The conventional no-slip boundary condition leads to a non-integrable stress singularity at a moving contact line. This makes numerical simulations challenging, especially when capillary effects are essential for the dynamics of the flow. This paper presents a new boundary methodology, suitable for numerical simulation of flow of two immiscible and incompressible fluids in the presence of moving contact points. The methodology is based on combining a relation between the apparent contact angle and the contact point velocity with the similarity solution for Stokes flow at a planar interface. The relation between angle and velocity can be determined by theoretical arguments, or from simulations using a more detailed model. The approach here uses the phase field model in a micro domain, with physically relevant parameters for molecular diffusion and interface thickness. The methodology is used to formulate a new boundary condition for the velocity. Numerical results illustrate the usefulness.

We consider a multiscale strategy addressing the disparate scales in the Landau-Lifschitz equations in micro-magnetism. At the microscopic scale, the dynamics of magnetic moments are driven by a high frequency field. On the macroscopic scale we are interested in simulating the dynamics of the magnetisation without fully resolving the microscopic scales. The method follows the framework of heterogeneous multiscale methods and it has two main ingredients: a micro- and a macroscale model. The microscopic model is assumed to be known exactly whereas the macro model is incomplete as it lacks effective quantities. The two models use different temporal and spatial scales and effective parameter values for the macro model are computed on the fly, allowing for improved efficiency over traditional one-scale schemes. For the analysis, we consider a single spin under a high frequency field and show that effective quantities can be obtained accurately with step-sizes much larger than the size of the microscopic scales required to resolve the microscopic features. Numerical results both for a single magnetic particle as well as a chain of interacting magnetic particles are given to validate the theory., 23 pages

A main challenge in numerical simulations of moving contact line problems is that the adherence, or no-slip boundary condition leads to a non-integrable stress singularity at the contact line. In this report we perform the first steps in developing the macroscopic part of an accurate multiscale model for a moving contact line problem in two space dimensions. We assume that a micro model has been used to determine a relation between the contact angle and the contact line velocity. An intermediate region is introduced where an analytical expression for the velocity exists. This expression is used to implement boundary conditions for the moving contact line at a macroscopic scale, along a fictitious boundary located a small distance away from the physical boundary. Model problems where the shape of the interface is constant thought the simulation are introduced. For these problems, experiments show that the errors in the resulting contact line velocities converge with the grid size $h$ at a rate of convergence $p\approx 2$.

We compare two high order finite-difference methods that solve the elastic wave equation in two dimensional domains with curved boundaries and material discontinuities. Two numerical experiments are designed with focus on wave boundary interaction, the response of a pressure wave impinging on a circular cavity and the wave field generated by an explosive impact on the wall an underground tunnel. Qualitative comparisons of the results are made where similarities and differences are pointed out., Comment: 12 pages, 5 figures

A recent study shows that the classical theory concerning accuracy and points per wavelength is not valid for surface waves in almost incompressible elastic materials. The grid size must instead be proportional to $(\frac{\mu}{\lambda})^{(1/p)}$ to achieve a certain accuracy. Here $p$ is the order of accuracy the scheme and $\mu$ and $\lambda$ are the Lame parameters. This accuracy requirement becomes very restrictive close to the incompressible limit where $\frac{\mu}{\lambda} \ll 1$, especially for low order methods. We present results concerning how to choose the number of grid points for 4th, 6th and 8th order summation-by-parts finite difference schemes. The result is applied to Lambs problem in an almost incompressible material., Comment: 14 pages, 5 figures

In this paper we study the stability of the perfectly matched layer (PML) for the elastic wave equation in rst order form. The theory of temporal stability of initial value problems corresponding to the PML is well developed. For initial boundary value problems (IBVPs) the theory of temporal stability of the PML is less complete. First, we study the solutions of two IBVPs corresponding to the PML for the elastic wave equation in rst order form. We consider separately a PML on the lower half-plane with free-surface boundary conditions at y = 0 and a PML on the left half-plane with characteristic boundary conditions at x = . In both cases the PML truncates a boundary in the x{direction. Using normal mode analysis we prove that the lower half{plane problem and the left half{plane problem do not support temporally growing modes. Second, we develop a high order accurate nite di erence approximation of the PML subject to the boundary conditions. To enable accurate and stable boundary treatments for the PML we construct continuous energy estimates in the Laplace space. We use summation-by-parts nite di erence operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. By mimicking the continuous energy estimates in the discrete setting, we construct stable numerical boundary procedures for the PML subject to the free-surface and the characteristic boundary conditions. Numerical experiments are presented corroborating the theoretical results.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia Química, Florianópolis, 2015. A modelagem de problemas onde a fronteira se modifica constantemente com o tempo pode se tornar desafiadora à medida que a malha tenha a necessidade de se adaptar constantemente. Nesse contexto, métodos computacionais onde a malha não se conforma com a fronteira são de grande interesse. Este trabalho propõe uma abordagem com o Método de Elementos Cortados para resolver equações diferenciais parciais utilizando malhas não alinhadas com o Método de Elementos Finitos. Como resultado da implementação proposta, foi desenvolvido o programa fem-cut-cell-3D, baseado na implementação em elementos finitos pela biblioteca deal.ii. A fim de avaliar matematicamente a implementação, quatro experimentos numéricos foram propostos: o problema clássico de Poisson, em duas e três dimensões; o problema de difusão de Laplace-Beltrami, em duas dimensões; e um caso transiente em duas dimensões de reação-difusão. Efeitos de estabilização da matriz de rigidez foram estudados para o problema de Poisson e Laplace-Beltrami em 2D, e a dependência teórica do número condicional com o tamanho dos elementos foi confirmada. Além disso, um parâmetro ótimo de estabilização foi definido. Taxas de convergência foram calculadas para os três primeiros casos e a estimativa teórica foi confirmada. Abstract : The modeling of problems where the boundary changes significantly over time may become challenging as the mesh needs to be adapted constantly. In this context, computational methods where the mesh does not conform to the boundary are of great interest. This paper proposes a stabilized cut-cell approach to solve partial differential equations using unfitted meshes using the Finite Element Method. As a result of the implementation of the method, the software fem-cut-cell-3D was developed, based on the implementation of the finite element method by the open source library deal.ii. In order to mathematically evaluate the method, four problems were proposed: the classical Poisson problem, in two and three dimensions; a pure diffusion Laplace-Beltrami problem, in two dimensions; and a reaction diffusion case in two dimensions. Stabilization effects on the stiffness matrix were studied for the Poisson and Laplace-Beltrami problems in 2D, and the theoretical dependence of the condition number with mesh size was confirmed. In addition, an optimal stabilization parameter was defined. Optimal convergence rates were obtained for the first three test cases.

A new fully adaptive multiresolution method is applied for the simulation of the complex dynamics of waves in excitable media in electrocardiology, where the membrane kinetics are given by the Aliev–Panfilov or Luo–Rudy II models. Numerical experiments show that the automatical adaptation strategy tracks the spatio-temporal pattern accurately at a substantially reduced computational cost if compared with fine-grid simulations. The nonlinear dynamics of complex multiscale patterns can thus be computed efficiently, also in the chaotic and turbulent regime which are currently beyond the frontiers of methods using regular discretizations.