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We present partially penalized immersed finite element methods for solving parabolic interface problems on Cartesian meshes. Typical semi-discrete and fully discrete schemes are discussed. Error estimates in an energy norm are derived. Numerical examples are provided to support theoretical analysis.

The aim of this paper is to propose a systematic way to obtain convergent finite element schemes for the Darcy-Stokes flow problem by combining well-known mixed finite elements that are separately convergent for Darcy and Stokes problems. In the approach in which the Darcy problem is set in its natural $\mathbf{H}(\text{div})$ formulation and the Stokes problem is expressed in velocity-pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the space of global velocity field. Our analysis of the Galerkin discretization of the resulting problem reveals that, if the mixed finite element space used in the Darcy domain admits an $\mathbf{H}(\text{div})$-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed finite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests confirming our theoretical convergence estimates.

Finite element methods are applied to the problem of characterizing linear, anisotropic elastic solids. The conventional finite element displacement formulation is used to simulate an elastic material in place stress. An inverted finite element formulation is then applied, and the characterizing six material constants are calculated as numerical results. A possible test device for the experimental characterization of anisotropic solids is postulated, the precision of displacement measurements to be required for such a device being determined by random perturbation analysis. Numerical constants accurate to within three percent are predicted if a precision of one part in eight hundred (1/800) can be measured. Numerical constants accurate to within one percent are predicted if a precision of one part in eight thousand (1/8000) can be measured in the test device.

It is well-known that it is comparatively difficult to design nonconforming finite elements on quadrilateral meshes by using Gauss-Legendre points on each edge of triangulations. One reason lies in that these degrees of freedom associated to these Gauss-Legendre points are not all linearly independent for usual expected polynomial spaces, which explains why only several lower order nonconforming quadrilateral finite elements can be found in literature. The present paper proposes two families of nonconforming finite elements of any odd order and one family of nonconforming finite elements of any even order on quadrilateral meshes. Degrees of freedom are given for these elements, which are proved to be well-defined for their corresponding shape function spaces in a unifying way. These elements generalize three lower order nonconforming finite elements on quadrilaterals to any order. In addition, these nonconforming finite element spaces are shown to be full spaces which is somehow not discussed for nonconforming finite elements in literature before.

The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in $L^\infty(\Omega)$. This class of coefficients includes as examples media with micro-structure as well as media with multiple non-separated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in \cite{107}, and elaborated in \cite{102}, \cite{103} and \cite{104}. The GFEM is constructed by partitioning the computational domain $\Omega$ into to a collection of preselected subsets $\omega_{i},i=1,2,..m$ and constructing finite dimensional approximation spaces $\Psi_{i}$ over each subset using local information. The notion of the Kolmogorov $n$-width is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom $N_{i}$ in the energy norm over $\omega_i$. The local spaces $% \Psi_{i}$ are used within the GFEM scheme to produce a finite dimensional subspace $S^N$ of $H^{1}(\Omega)$ which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially (i.e., super-algebraicly) with respect to the degrees of freedom $N$. When length scales "`separate" and the microstructure is sufficiently fine with respect to the length scale of the domain $\omega_i$ it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the pre-asymtotic regime.

Over the past few years finite element based procedures for the solution of high speed viscous compressible flows were developed. The objective of this research is to build upon the finite element concepts which have already been demonstrated and to develop these ideas to produce a method which is applicable to the solution of large scale practical problems. The problems of interest range from three dimensional full vehicle Euler simulations to local analysis of three-dimensional viscous laminar flow. Transient Euler flow simulations involving moving bodies are also to be included. An important feature of the research is to be the coupling of the flow solution methods with thermal/structural modeling techniques to provide an integrated fluid/thermal/structural modeling capability. The progress made towards achieving these goals during the first twelve month period of the research is presented.

Non-overlapping domain decomposition methods necessarily have to exchange Dirichlet and Neumann traces at interfaces in order to be able to converge to the underlying mono-domain solution. Well known such non-overlapping methods are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and optimized Schwarz methods. For all these methods, cross-points in the domain decomposition configuration where more than two subdomains meet do not pose any problem at the continuous level, but care must be taken when the methods are discretized. We show in this paper two possible approaches for the consistent discretization of Neumann conditions at cross-points in a Finite Element setting.

High-order finite element methods (also known as spectral/hp element methods) using either the continuous Galerkin or discontinuous Galerkin formulation have reached a level of sophistication such that they are now commonly applied to a diverse set of real-life engineering problems. Visualization of computed results is often used as a means of understanding and evaluating the numerical approximation of the mathematical model, and it provides a means of closing the loop - that is, of critically evaluating the computational results for refinement of the model and/or numerics or for interpretation of the physical world. Visualizations of high-order finite element results which do not respect the a priori knowledge of how the data were produced and which do not provide a quantification of the visual error produced undermine the scientific process just described. The goals of this effort are to define, investigate, and address the technical obstacles inherent in visualization of data derived from high-order numerical methods and to develop algorithms and software solutions that can be employed by the high-order simulation community.

In this short note we investigate the numerical performance of the method of artificial diffusion for second-order fully nonlinear Hamilton-Jacobi-Bellman equations. The method was proposed in (M. Jensen and I. Smears, arxiv:1111.5423); where a framework of finite element methods for Hamilton-Jacobi-Bellman equations was studied theoretically. The numerical examples in this note study how the artificial diffusion is activated in regions of degeneracy, the effect of a locally selected diffusion parameter on the observed numerical dissipation and the solution of second-order fully nonlinear equations on irregular geometries.

A computational method for accurately predicting the static aeroelastic deformations of typical transonic transport wind tunnel models is described. The method utilizes a finite element method (FEM) for predicting the deformations. Extensive calibration/validation of this method was carried out using a novel wind-off wind tunnel model static loading experiment and wind-on optical wing twist measurements obtained during a recent wind tunnel test in the National Transonic Facility (NTF) at NASA LaRC. Further validations were carried out using a Navier-Stokes computational fluid dynamics (CFD) flow solver to calculate wing pressure distributions about several aeroelastically deformed wings and comparing these predictions with NTF experimental data. Results from this aeroelastic deformation method are in good overall agreement with experimentally measured values. Including the predicted deformations significantly improves the correlation between CFD predicted and experimentally measured wing & pressures.

A new approach to a posteriori error estimation is outlined which is applicable to general h-p finite element approximations of general classes of boundary value problems. The approach makes use of duality arguments and is based on the element residual method (ERM). Important aspects of the method are that it provides a systematic approach toward deriving element boundary conditions for the ERM; it leads to an upper bound for the global error in an appropriate energy norm; and it is valid for non-uniform and irregular h-p meshes. In the present exposition, a brief outline of the theoretical foundations of the method is given together with the results of its application to several representative problems. These results show that the approach is applicable to general linearly elliptic systems, including unsymmetrical operators, and that the method is valid for broad classes of linear and non- linear problems.

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

An overview on recent applications of the finite-element method Maxwell-solver JCMsuite to simulation tasks in nanooptics is given. Numerical achievements in the fields of optical metamaterials, plasmonics, photonic crystal fibers, light emitting devices, solar cells, optical lithography, optical metrology, integrated optics, and photonic crystals are summarized.

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal order error estimates in both H^1 and L^2 norms are established. Numerical tests are performed and reported.

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods, but also are of a great potential to be useful in error analysis for other IFE methods.

Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales (see Figure 1 for the illustration of a perforated domain). Moreover, these problems are intrinsically multiscale and their discretizations can yield very large linear or nonlinear systems. In this paper, we investigate multiscale approaches that attempt to solve such problems on a coarse grid by constructing multiscale basis functions in each coarse grid, where the coarse grid can contain many perforations. In particular, we are interested in cases when there is no scale separation and the perforations can have different sizes. In this regard, we mention some earlier pioneering works [14, 18, 17], where the authors develop multiscale finite element methods. In our paper, we follow Generalized Multiscale Finite Element Method (GMsFEM) and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems. We show that with a few basis functions in each coarse block, one can accurately approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes equations in perforated domain. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere.

Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales (see Figure 1 for the illustration of a perforated domain). Moreover, these problems are intrinsically multiscale and their discretizations can yield very large linear or nonlinear systems. In this paper, we investigate multiscale approaches that attempt to solve such problems on a coarse grid by constructing multiscale basis functions in each coarse grid, where the coarse grid can contain many perforations. In particular, we are interested in cases when there is no scale separation and the perforations can have different sizes. In this regard, we mention some earlier pioneering works [14, 18, 17], where the authors develop multiscale finite element methods. In our paper, we follow Generalized Multiscale Finite Element Method (GMsFEM) and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems. We show that with a few basis functions in each coarse block, one can accurately approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes equations in perforated domain. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere.

We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution.

A recently developed spectral-element adaptive refinement incompressible magnetohydrodynamic (MHD) code [Rosenberg, Fournier, Fischer, Pouquet, J. Comp. Phys. 215, 59-80 (2006)] is applied to simulate the problem of MHD island coalescence instability (MICI) in two dimensions. MICI is a fundamental MHD process that can produce sharp current layers and subsequent reconnection and heating in a high-Lundquist number plasma such as the solar corona [Ng and Bhattacharjee, Phys. Plasmas, 5, 4028 (1998)]. Due to the formation of thin current layers, it is highly desirable to use adaptively or statically refined grids to resolve them, and to maintain accuracy at the same time. The output of the spectral-element static adaptive refinement simulations are compared with simulations using a finite difference method on the same refinement grids, and both methods are compared to pseudo-spectral simulations with uniform grids as baselines. It is shown that with the statically refined grids roughly scaling linearly with effective resolution, spectral element runs can maintain accuracy significantly higher than that of the finite difference runs, in some cases achieving close to full spectral accuracy.

This paper studies finite element approximations of the stochastic Allen-Cahn equation with gradient-type multiplicative noises that are white in time and correlated in space. The sharp interface limit as the parameter $\epsilon \rightarrow 0$ of the stochastic equation formally approximates a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Both the stochastic Allen-Cahn equation and the stochastic mean curvature flow arise from materials science, fluid mechanics and cell biology applications. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with a crucial help of the H\"{o}lder continuity in time with respect to the spatial $L^2$-norm and $H^1$-seminorm for the strong solution of the stochastic Allen-Cahn equation, which are key technical lemmas proved in paper. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temporal norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noises.

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no "sufficiently large" parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are given to support the theoretical results.

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau--Lifshitz--Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. \emph{A priori} energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both \emph{a priori} energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank--Nicolson method. The latter is solved by using the former as a non-linear solver.

We survey some mathematical aspects of finite element methods for incompressible viscous flows, concentrating on the steady primitive variable formulation. We address the discretization of a weak formulation of the Navier Stokes equations; we then consider the div-stability condition, whose satisfaction insures the stability of the approximation. Specific choices of finite element spaces for the velocity and pressure are then discussed. Finally, the connection between different weak formulations and a variety of boundary conditions is explored.

Low Cycle Fatigue (LCF) in engine components involves macroscopic cyclic plastic strains (with a stress-strain hysteresis loop) over a significant portion of the failure region. Characterising elasto-plastic behaviour in potential failure regions is a necessary step in estimating LCF life. The equations governing elasto plastic behaviour are summarized, and the methods of implementing them in Finite Element (FE) stress analysis programs, discussed. An extension of the PAFEC program to include mixed isotropickinematic hardening is outlined, and verified by examples for which alternative FE solutions were available. A sample application has been made to holes in a plate with biaxial stress fields similar to those in disc webs, and the results compared with the Neuber and modified Stowell rules commonly used for design life estimation; these rules tend to overestimate the strain in biaxial stress conditions, leading to conservative life estimates.

The thrust of this work was the development of efficient and accurate finite element methods for flow problems. Specific applications include periodic acoustic problems, potential flow problems and incompressible viscous flows. However, the theoretical analyses carried out also have a direct bearing on the approximation of problems in other areas, e.g., electromagnetics and elasticity. For the particular fluids applications mentioned above, computer codes implementing the algorithms have also been developed.

Finite element methods to compute approximate solutions to flow problems involving the flows of viscoelastic fluids are discussed. The primary goals of such investigations are at least three: First, to evaluate the predictions of the many proposed constitutive theories for viscoelastic fluids. Second, to model measurement flows in various rheological measurement devices in order to quantify the deviation of the actual flow from the flow which must be presumed to interpret the measurement. Third, it is hoped that these methods will prove sufficiently robust to allow the simulation of idealized polymer processes with the aim of aiding in the design of such processes and the required apparatus. The focus of the current research of the author and a growing number of others is on two-dimensional, isothermal, steady flows of incompressible fluids. While these restrictions will be seen to be non-essential in theory, even the simplest calculations of non-visometric flow solutions will be seen to require a high degree of computational complexity in practice. Nevertheless, the current finite element procedures seem to show promise in the continuing endeavor to understand this challenging class of problems. Keywords: Constitutive equation; Viscoelastic fluid; Finite element; Measurement flow; Differential model; Integral model; Characteristics; Streamlines; Hole-pressure.

Finite element methods to compute approximate solutions to flow problems involving the flows of viscoelastic fluids are discussed. The primary goals of such investigations are at least three: First, to evaluate the predictions of the many proposed constitutive theories for viscoelastic fluids. Second, to model measurement flows in various rheological measurement devices in order to quantify the deviation of the actual flow from the flow which must be presumed to interpret the measurement. Third, it is hoped that these methods will prove sufficiently robust to allow the simulation of idealized polymer processes with the aim of aiding in the design of such processes and the required apparatus. The focus of the current research of the author and a growing number of others is on two-dimensional, isothermal, steady flows of incompressible fluids. While these restrictions will be seen to be non-essential in theory, even the simplest calculations of non-visometric flow solutions will be seen to require a high degree of computational complexity in practice. Nevertheless, the current finite element procedures seem to show promise in the continuing endeavor to understand this challenging class of problems. Keywords: Constitutive equation; Viscoelastic fluid; Finite element; Measurement flow; Differential model; Integral model; Characteristics; Streamlines; Hole-pressure.

The Adaptive Hydraulics (ADH) model is an adaptive finite element method to simulate three-dimensional Navier-Stokes flow, unsaturated and saturated groundwater flow, overland flow, and two- or three-dimensional shallow-water flow and transport. In the shallow-water flow and transport, especially involving multispecies transport, the water depth (h), the product of water depth and velocities (uh and vh), as well as water depth and chemical concentration (hc) are dependent variables of fluid-motion simulations and are often solved at various times. It is important for the numerical model to predict accurate water depth, velocity fields, and chemical distribution, as well as conserve mass, especially for water quality applications. Solution accuracy depends highly on mesh resolution. Adaptive mesh refinement (AMR), particularly the h-refinement, is often used to add new nodes in the region where they are needed and to remove others where they are no longer required during the simulation. The AMR is proven to optimize the performance of a computed solution. However, mass with gain or loss can occur when elements are merged due to removing a node at mesh coarsening. Therefore, we develop and implement the mass-conservative unrefinement algorithm to ensure the mass conserved in a merged element in which a node has been removed. This study describes the use of the Galerkin finite element method to redistribute mass to nodes comprising a merged element. The algorithm was incorporated into the ADH code. This algorithm minimizes mass error during the unrefinement process to conserve mass during the simulation for two-dimensional shallow-water flow and transport. The implementation neither significantly increases the computational time nor memory usage. The simulation was run with various numbers of processors. The results showed good scaling of solution time as the number of processors increases.

This study investigates the time domain coupling of the boundary and finite element methods for elastodynamics. The study is motivated by the need for a 'silent boundary' in nonlinear problems with infinite domains. Of the boundary element methods (BEMs) considered, formulations based on the Stokes solution appear to be the best suited. The theoretical and numerical bases for the indirect and direct BEMs using the Stokes solution are presented. The integral equation statements of the boundary-initial value problem included a convolution of the time variable and thus appear to be computationally intense. A coupling algorithm which treats the BEM region as a nonlinear boundary condition to the finite element region is presented. The potential of the coupled approach might be realized by exploiting special properties of Stoke's solution and designing the algorithm to use the advances in computer hardware (e.g. parallel processing).

Elementary descriptions of finite element and finite difference methods are given while the finite volume method is briefly overviewed. Examples illustrating finite element and finite difference methods are worked out. Finally, comparisons of these methods between themselves and with some examples from literature are given. (AN)

At the start, three and a half years ago, the finite fluid element method (Section 1) was the only one under consideration. Things went badly with that method, and progress with its implementation went far more slowly than we ever anticipated. As a result, by the end of the first year, a second method, which originally was developed for a check on results of the first, had become by far the more promising of the two. Most of this report (Sections 3-6) describes progress we have made with the application of biased differences (Section 2) to a variety of fairly difficult problems of numerical fluid mechanics. Finally, in the last six months of the period covered by this report, the major difficulties with finite fluid elements were overcome, so it became possible to begin a comparison of the two methods (Section 6). Preliminary indications are that both methods are reliable, and both are considerably more efficient than a third method with which they have been compared. Section 7 is a report of progress with a boundary integral method that is not closely related to the others except by being numerical, and Section 8 is a description of the kinds of graphical software we had to develop for interpretation of our numerical computations.

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

A rational function is presented for approximating the function f(z) = coth z - 1/z that appears in several exponentially fitted or weighted finite difference and finite element methods for convection-diffusion problems. The approximation is less expensive to evaluate than f(z) and provides greater accuracy than the doubly asymptotic approximation when z = 0(1).

GAMES103：基于物理的计算机动画入门 主讲老师：王华民 俄亥俄州立大学，凌迪科技首席科学家 课程主页：http://games-cn.org/games103/ 更多计算机图形学相关的学习资料请访问： 计算机图形学与混合现实在线平台GAMES: http://games-cn.org

Modern numerical methods for the optimization of large discretized systems are now well developed and highly efficient in the case of thin walled elastic structures modeled by finite elements. However, this is not yet true for structures whose components are subject simultaneously to bending and extension loads. In this paper, the idea of Generalized Optimality Criterion (GOC), set forth in previous final scientific reports for bar, membrane and pure bending elements, is extended to deal with general beam and flat shell elements. The modifications brought to the GOC result in explicit approximations for the behavior constraints.(Author)

Dirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

Dirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

In a previously reported investigation, scattered sound pressures and velocity distributions on the surfaces of submerged infinite cylinders ensonified by steady-state plane waves were calculated. The numerical method used for these calculations was based on a finite element formulation of the coupled fluid-structure interaction problem and implemented using the NASTRAN computer program. In the present investigation, these pressure and velocity data are used as the basis for numerical calculations of the scattered sound pressures generated in the exterior fields of the cylinders. The method consists of replacing an infinite cylinder by a finite cylinder whose external sound field, in three dimensions, converges to the required two-dimensional (2-D) field as the cylinder length increases. The field pressures are obtained by numerical quadrature of the HELMHOLTZ integral formulation which relates external field pressures to pressures and normal velocities at the structural surface. Calculated 2-D pressure fields, obtained for rigid and elastic cylinders, are in excellent agreement with analytical results. (Author)

In a previously reported investigation, scattered sound pressures and velocity distributions on the surfaces of submerged infinite cylinders ensonified by steady-state plane waves were calculated. The numerical method used for these calculations was based on a finite element formulation of the coupled fluid-structure interaction problem and implemented using the NASTRAN computer program. In the present investigation, these pressure and velocity data are used as the basis for numerical calculations of the scattered sound pressures generated in the exterior fields of the cylinders. The method consists of replacing an infinite cylinder by a finite cylinder whose external sound field, in three dimensions, converges to the required two-dimensional (2-D) field as the cylinder length increases. The field pressures are obtained by numerical quadrature of the HELMHOLTZ integral formulation which relates external field pressures to pressures and normal velocities at the structural surface. Calculated 2-D pressure fields, obtained for rigid and elastic cylinders, are in excellent agreement with analytical results. (Author)

In a previously reported investigation, scattered sound pressures and velocity distributions on the surfaces of submerged infinite cylinders ensonified by steady-state plane waves were calculated. The numerical method used for these calculations was based on a finite element formulation of the coupled fluid-structure interaction problem and implemented using the NASTRAN computer program. In the present investigation, these pressure and velocity data are used as the basis for numerical calculations of the scattered sound pressures generated in the exterior fields of the cylinders. The method consists of replacing an infinite cylinder by a finite cylinder whose external sound field, in three dimensions, converges to the required two-dimensional (2-D) field as the cylinder length increases. The field pressures are obtained by numerical quadrature of the HELMHOLTZ integral formulation which relates external field pressures to pressures and normal velocities at the structural surface. Calculated 2-D pressure fields, obtained for rigid and elastic cylinders, are in excellent agreement with analytical results. (Author)

In a previously reported investigation, scattered sound pressures and velocity distributions on the surfaces of submerged infinite cylinders ensonified by steady-state plane waves were calculated. The numerical method used for these calculations was based on a finite element formulation of the coupled fluid-structure interaction problem and implemented using the NASTRAN computer program. In the present investigation, these pressure and velocity data are used as the basis for numerical calculations of the scattered sound pressures generated in the exterior fields of the cylinders. The method consists of replacing an infinite cylinder by a finite cylinder whose external sound field, in three dimensions, converges to the required two-dimensional (2-D) field as the cylinder length increases. The field pressures are obtained by numerical quadrature of the HELMHOLTZ integral formulation which relates external field pressures to pressures and normal velocities at the structural surface. Calculated 2-D pressure fields, obtained for rigid and elastic cylinders, are in excellent agreement with analytical results. (Author)

In a previously reported investigation, scattered sound pressures and velocity distributions on the surfaces of submerged infinite cylinders ensonified by steady-state plane waves were calculated. The numerical method used for these calculations was based on a finite element formulation of the coupled fluid-structure interaction problem and implemented using the NASTRAN computer program. In the present investigation, these pressure and velocity data are used as the basis for numerical calculations of the scattered sound pressures generated in the exterior fields of the cylinders. The method consists of replacing an infinite cylinder by a finite cylinder whose external sound field, in three dimensions, converges to the required two-dimensional (2-D) field as the cylinder length increases. The field pressures are obtained by numerical quadrature of the HELMHOLTZ integral formulation which relates external field pressures to pressures and normal velocities at the structural surface. Calculated 2-D pressure fields, obtained for rigid and elastic cylinders, are in excellent agreement with analytical results. (Author)

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The goal of this paper is to develop and analyze some fully discrete finite element methods for a displacement-pressure model modeling swelling dynamics of polymer gels under mechanical constraints. In the model, the swelling dynamics is governed by the solvent permeation and the elastic interaction; the permeation is described by a pressure equation for the solvent, and the elastic interaction is described by displacement equations for the solid network of the gel. By introducing an "elastic pressure" we first present a reformulation of the original model, and then propose a time-stepping scheme which decouples the PDE system at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field and another is a diffusion problem for a "pseudo-pressure" field. To make such a multiphysical approach feasible, it is vital to discover admissible constraints to resolve the uniqueness issue for both sub-problems. The main advantage of the proposed approach is that it allows one to utilize any convergent Stokes solver together with any convergent diffusion equation solver to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are used as an example to present the ideas and to demonstrate the viability of the proposed multiphysical approach. It is proved that, under a mesh constraint, both the proposed semi-discrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for the numerical solutions of both the semi-discrete and fully discrete methods. Numerical experiments are also presented to show the efficiency of the proposed approach and methods.

This paper develops and analyzes some continuous interior penalty finite element methods (CIP-FEMs) using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. The novelty of the proposed methods is to use complex penalty parameters with positive imaginary parts. It is proved that, if the penalty parameter is a pure imaginary number $\i\ga$ with $0 C_0$ and $kh$ is bounded, where $k$ is the wave number, $h$ is the mesh size, and the $C$'s are positive constants independent of $k$, $h$, and $\ga$. Optimal order $L^2$ error estimates are also derived. The analysis is also applied if the penalty parameter is a complex number with positive imaginary part. By taking $\ga\to 0+$, the above estimates are extended to the linear finite element method under the condition $k^3h^2\le C_0$. Numerical results are provided to verify the theoretical findings. It is shown that the penalty parameters may be tuned to greatly reduce the pollution errors.

An efficient time-stepping algorithm is proposed based on operator-splitting and the space-time discontinuous Galerkin finite element method for problems in the non-classical theory of thermoelasticity. The non-classical theory incorporates three models; the classical theory based on Fourier's law of heat conduction resulting in a hyperbolic-parabolic coupled system, a non-classical theory of a fully hyperbolic extension, and a combination of the two. The general problem is split into two contractive sub-problems, namely the mechanical phase and the thermal phase. Each sub-problem is discretised using space-time discontinuous Galerkin finite element method resulting each to be stable which then leads to unconditional stability of the global product algorithm. A number of numerical examples are presented to demonstrate the performance and capability of the method.

We introduce a framework for the design of finite element methods for two-dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy, both in space and in time. At the core of our approach is the use of a universal mesh: a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary. The resulting method maintains an exact representation of the (prescribed) moving boundary at the discrete level, or an approximation of the appropriate order, yet is immune to large distortions of the mesh under large deformations of the domain. The framework is general, making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite-element space on the universal mesh for the problem at hand, and a preferred and suitable time integrator for ordinary differential equations. We illustrate our approach by constructing a particular class of methods, and apply them to a prescribed-boundary variant of the Stefan problem. We present numerical evidence for the order of accuracy of our schemes in one and two dimensions.

Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.