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The authors discuss a finite element method for solving initial-boundary value problems for vector systems of partial differential equations in one space dimension and time. The method automatically adjusts the computational mesh as the solution evolves in time so as to approximately minimize the local discretization error. They are thus able to calculate accurate solutions with fewer elements than would be necessary with a uniform mesh. This overall method contains two distinct steps: a solution step and a mesh selection step. They solve the partial differential equations using a finite element-Galerkin method on trapezoidal space-time-elements with either piecewise linear or cubic Hermits polynomial approximations. A variety of mesh selection strategies are discussed and analyzed. Results are presented for several computational examples.

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)

The general goals of this project included the development of new, high-order, adaptive methods for the computer simulation of stress-wave propagation phenomena in solid mechanics, particularly elastodynamics. The motivation for using adaptive methodologies is to control and optimize the computational process, to use a posteriori error estimates to optimize meshes and spectral orders of approximation. Such orchestrated meshing can produce exponential rates of convergence, thereby allowing complex simulations to be done using orders-of-magnitude fewer unknowns than standard methods.

Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantage of adaptive strategies versus non-adaptive strategies.

The primary purpose of this paper is to compare the accuracy and performance of two numerical approaches to solving systems of partial differential equations. These equations are posed on adjoining domains sharing boundary conditions on a common boundary interface in the important case when the meshes used on the two domains are non-matching across the interface. The first widely used approach is based on a finite volume method employing ad hoc projections to relate approximations on the two domains across the interface. The second approach uses the mathematically-founded mortar mixed finite element method. To quantify the performance, we use a goal-oriented a-posteriori error estimate that quantifies various aspects of discretization error to the overall error. While the performance difference may be not a surprise in some cases, we believe that there is a perception in part of the scientific community concerned with multiphysics systems that if the solution is smooth near the interface, then there is little effect from varying the coupling technique. We find that, on the contrary, the error associated with ad hoc coupling approaches may be large in practical situations. Moreover, we also show that mortar methods can be used with black box component solves, thus permitting an efficient and practical implementation also within legacy codes.

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.

The applicants have developed a finite element based approach for the solution of three-dimensional compressible flows. The procedure enables flow solutions to be obtained on tetrahedral discretizations of computational domains of complex form. A further development was the incorporation of a solution adaptive mesh strategy in which the adaptivity is achieved by complete remeshing of the solution domain. During the previous year, the applicants were working with the Advanced Aerodynamics Concepts Branch at NASA Ames Research Center with an implementation of the basic meshing and solution procedure. The objective of the work to be performed over this twelve month period was the transfer of the adaptive mesh technology and also the undertaking of basic research into alternative flow algorithms for the Euler equations on unstructured meshes.

A general numerical strategy for solving partial differential equations and other functional problems by cycling between coarser and finer levels of discretization is described. Optimal discretization schemes are provided together with very fast general solvers. It is described in terms of finite element discretizations of general nonlinear minimization problems. The basic processes (relaxation sweeps, fine-grid-to-coarse-grid transfers of residuals, coarse-to-fine interpolations of corrections) are directly and naturally determined by the objective functional and the sequence of approximation spaces. The natural processes, however, are not always optimal. Concrete examples are given and some new techniques are reviewed. Including the local truncation extrapolation and a multilevel procedure for inexpensively solving chains of many boundary value problems, such as those arising in the solution of time-dependent problems.

In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.

The weak Galerkin (WG) methods have been introduced in the references [11, 16] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique.

We analyze the flux conservation property of the finite element method. It is shown that the finite element solution does approximate the flux locally in the optimal order, i.e., the same order as that of the nodal interpolation operator. We propose two methods, post-processing the finite element solutions locally. The new solutions, remaining as optimal-order solutions, are flux-conserving elementwise. In one of our methods, the processed solution also satisfies the original finite element equations. While the high-order finite volume schemes are still under construction, our methods produce finite-volume-like finite element solution of any order. In particular, our methods avoid solving non-symmetric finite volume equations. Numerical tests in 2D and 3D verify our findings.

We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity and Lagrange multipliers. We use multiscale basis functions for the both velocity and gradient of pressure. In the expanded mixed MsFEM framework, we consider both cases of separable-scale and non-separable spatial scales. We specifically analyze the methods in three categories: periodic separable scales, $G$- convergence separable scales, and continuum scales. When there is no scale separation, using some global information can improve accuracy for the expanded mixed MsFEMs. We present rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes both conforming and nonconforming expanded mixed MsFEM. Numerical results are presented for various multiscale models and flows in porous media with shales to illustrate the efficiency of the expanded mixed MsFEMs.

Aim: The most common type of distal femur physis fractures is Salter-Harris type 2 (SH type 2). These fractures have high complication rates and can cause a significant loss of function. Anatomical reduction is important in treatment. In this study, we aimed to compare two methods commonly used in SH type 2 fractures, cross pinning, and two parallel screw fixation methods using the finite element method. Material and Methods: The SH type 2 fracture model was created in the femur model obtained from the 3-dimension (3D) computed tomography (CT) scans. The fracture in the first model was fixed with crossed Kirschner (K) wires. The fracture in the second model was fixed with two parallel screws placed from the metaphyseal part. The two models created were moved to the Ansys Workbench program. Axial overload, varus, valgus, anterior, posterior bending, and torsional forces were applied and analyzed with the 3D finite element method. Results: In axial overload, the max stress in growth cartilage K wire was 0.40 MPa, while in the screw- 1.24 MPa. The varus bending was 0.32 MPa and 1.71 MPa, respectively. Also, the valgus bending was 0.15 MPa and 0.56 MPa, respectively. The anterior bending was 0.85 MPa and 1.30 MPa, respectively. Also, the posterior bending was 0.56 MPa and 2.01 MPa, respectively When torsional force was applied, it was found as 0.008 MPa and 0.16 MPa, respectively. Discussion: In SH type 2 distal femoral fractures, the cross-pinning method is superior to the two parallel screw methods placed from the metaphyseal part in bending, torsion and axial loads.

Aim: The most common type of distal femur physis fractures is Salter-Harris type 2 (SH type 2). These fractures have high complication rates and can cause a significant loss of function. Anatomical reduction is important in treatment. In this study, we aimed to compare two methods commonly used in SH type 2 fractures, cross pinning, and two parallel screw fixation methods using the finite element method. Material and Methods: The SH type 2 fracture model was created in the femur model obtained from the 3-dimension (3D) computed tomography (CT) scans. The fracture in the first model was fixed with crossed Kirschner (K) wires. The fracture in the second model was fixed with two parallel screws placed from the metaphyseal part. The two models created were moved to the Ansys Workbench program. Axial overload, varus, valgus, anterior, posterior bending, and torsional forces were applied and analyzed with the 3D finite element method. Results: In axial overload, the max stress in growth cartilage K wire was 0.40 MPa, while in the screw- 1.24 MPa. The varus bending was 0.32 MPa and 1.71 MPa, respectively. Also, the valgus bending was 0.15 MPa and 0.56 MPa, respectively. The anterior bending was 0.85 MPa and 1.30 MPa, respectively. Also, the posterior bending was 0.56 MPa and 2.01 MPa, respectively When torsional force was applied, it was found as 0.008 MPa and 0.16 MPa, respectively. Discussion: In SH type 2 distal femoral fractures, the cross-pinning method is superior to the two parallel screw methods placed from the metaphyseal part in bending, torsion and axial loads.

We consider error estimates for the fully discretized instationary Navier-Stokes problem. For the spatial approximation we use conforming inf-sup stable finite element methods in conjunction with grad-div and local projection stabilization acting on the streamline derivative. For the temporal discretization a pressure-correction projection algorithm based on BDF2 is used. We can show quasi-optimal rates of convergence with respect to time and spatial discretization for all considered error measures. Some of the error estimates are quasi-robust with respect to the Reynolds number.

We consider error estimates for the fully discretized instationary Navier-Stokes problem. For the spatial approximation we use conforming inf-sup stable finite element methods in conjunction with grad-div and local projection stabilization acting on the streamline derivative. For the temporal discretization a pressure-correction projection algorithm based on BDF2 is used. We can show quasi-optimal rates of convergence with respect to time and spatial discretization for all considered error measures. Some of the error estimates are quasi-robust with respect to the Reynolds number.

This study examines various computational techniques to analyze dynamic response and failure of sandwich composite materials subject to fluid-structure interaction characterized by an acoustic field or the propagation of velocity potential according to the wave equation. A displacement-only plate finite element is developed and implemented using Discontinuous Galerkin (DG) methodology; its accuracy compares favorably to both theory and Continuous Galerkin methods. Several approaches to analyzing debonding failure between skin and core layers of sandwich com-posite structures are demonstrated and evaluated; partial disconnection between neighboring elements at a debonding site shows good qualitative agreement with known physical phenomena. A hybrid Finite Element-Cellular Automata (FE+CA) approach to modeling an acoustic field with non-reflecting boundary conditions is presented, validated nu-merically, and favorably compared with experimental results. The FE+CA fluid model is then combined with the DG structural model to simulate fluid-structure interaction; this combined model compared favorably with experimental results for the strain field of laminated plates subject to low-velocity impact. Each technique addressed shows promise for flexible and accurate modeling of failure initiation and propagation in sandwich and laminate composites subject to fluid-structure interaction with moderate computational costs.

In this paper, we present a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of oversampling techniques is to introduce a small dimensional snapshot space. We present numerical results for two-phase flow and transport, without updating basis functions in time. Our numerical results show that one can achieve good accuracy with a few basis functions per coarse edge if one selects appropriate offline spaces.

We propose a simple post-processing technique for linear and high order continuous Galerkin Finite Element Methods (CGFEMs) to obtain locally conservative flux field. The post-processing technique requires solving an auxiliary problem on each element independently which results in solving a linear algebra system whose size is low for any order CGFEM. The post-processing could have been done directly from the finite element solution that results in locally conservative flux on the element. However, the normal flux is not continuous at the elemental boundary. To construct locally conservative flux field whose normal component is also continuous, we propose to do the post-processing on the nodal-centered control volumes which are constructed from the original finite element mesh. We show that the post-processed solution converges in an optimal fashion to the true solution in an H1 semi-norm. We present various numerical examples to demonstrate the performance of the post-processing technique.

In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to represent the solution on a coarse grid with multiscale basis functions computed offline. Computing the GMsFEM solution involves calculating the residuals on the fine grid. We use empirical interpolation concepts to evaluate the residuals and the Jacobians of the multiscale system with a computational cost which is proportional to the coarse scale problem rather than the fully-resolved fine scale one. Empirical interpolation methods use basis functions and an inexpensive inversion which are computed in the offline stage for finding the coefficients in the expansion based on a limited number of nonlinear function evaluations. The proposed multiscale empirical interpolation techniques: (1) divide computing the nonlinear function into coarse regions; (2) evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced-order representation of the solution; and (3) introduce multiscale proper-orthogonal-decomposition techniques to find appropriate interpolation vectors. We demonstrate the effectiveness of the proposed methods on several examples of nonlinear multiscale PDEs that are solved with Newton's methods and fully-implicit time marching schemes. Our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale PDEs on a coarse grid with bounded error.

In the first part of this paper, we establish a conditional optimality result for an adaptive mixed finite element method for the stationary Stokes problem discretized by the standard Taylor-Hood elements, under the assumption of the so-called general quasi-orthogonality. Optimality is measured in terms of a modified approximation class defined through the total error, as is customary since the seminal work of Cascon, Kreuzer, Nochetto and Siebert. The second part of the paper is independent of optimality results, and concerns interrelations between the modified approximation classes and the standard approximation classes (the latter defined through the energy error). Building on the tools developed in the papers of Binev, Dahmen, DeVore, and Petrushev, and of Gaspoz and Morin, we prove that the modified approximation class coincides with the standard approximation class, modulo the assumption that the data is regular enough in an appropriate scale of Besov spaces.

ADA756513

In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying an ensemble of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.

We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy.

In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.

We develop a high order cut finite element method for the Stokes problem based on general inf-sup stable finite element spaces. We focus in particular on composite meshes consisting of one mesh that overlaps another. The method is based on a Nitsche formulation of the interface condition together with a stabilization term. Starting from inf-sup stable spaces on the two meshes, we prove that the resulting composite method is indeed inf-sup stable and as a consequence optimal \emph{a~priori} error estimates hold.

We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and $L^2$ norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

We propose finite element methods for compressible barotropic Stokes systems. We state convergence results for these methods and outline their proofs. The principal tools of the proofs are higher integrability estimates for the discrete density, equations for the discrete effective viscous flux, and renormalized formulations of the numerical method for the density equation.

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous piecewise polynomials on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in $\mathbb{R}^3$.

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.

In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order semilinear elliptic equations. We first introduce several approximate dual problems, and briefly discuss the target problem class. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for semilinear problems, and then describe goal-oriented variations of the standard approach to AFEM (GOAFEM). Following the recent approach of Mommer-Stevenson and Holst-Pollock for linear problems, we first establish a contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence in the sense of the quantity of interest. Numerical experiments support the theoretical results.

In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order semilinear elliptic equations. We first introduce several approximate dual problems, and briefly discuss the target problem class. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for semilinear problems, and then describe goal-oriented variations of the standard approach to AFEM (GOAFEM). Following the recent approach of Mommer-Stevenson and Holst-Pollock for linear problems, we first establish a contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence in the sense of the quantity of interest. Numerical experiments support the theoretical results.

Book Source: Digital Library of India Item 2015.193316 dc.contributor.author: Amiya Kumar Pani dc.date.accessioned: 2015-07-08T02:30:55Z dc.date.available: 2015-07-08T02:30:55Z dc.date.digitalpublicationdate: 2005-08-27 dc.identifier.barcode: 1990010086728 dc.identifier.origpath: /rawdataupload/upload/0086/728 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/193316 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 180 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.title: Finite Element Methods For Nonlinear Free Boundary Problems

Book Source: Digital Library of India Item 2015.193316 dc.contributor.author: Amiya Kumar Pani dc.date.accessioned: 2015-07-08T02:30:55Z dc.date.available: 2015-07-08T02:30:55Z dc.date.digitalpublicationdate: 2005-08-27 dc.identifier.barcode: 1990010086728 dc.identifier.origpath: /rawdataupload/upload/0086/728 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/193316 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 180 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Mechanical Engineering In General. Nuclear Technology. Electrical Engineering. Machinery dc.title: Finite Element Methods For Nonlinear Free Boundary Problems

A novel residual-type {\it a posteriori } error analysis technique is   developed for multipoint flux mixed finite element methods for flow in porous media   in two or three space dimensions. The derived {\it a posteriori } error estimator for  the velocity and pressure error   in $L^{2}-$norm  consists of discretization and quadrature indicators, and is shown to be   reliable and efficient. The main tools of analysis are   a   locally postprocessed approximation to the pressure solution of   an auxiliary problem and a quadrature   error estimate. Numerical experiments are presented to  illustrate   the competitive behavior of the estimator.

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.

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.

Average elastic Von Mises equivalent stresses were calculated along the throat of a single lap fillet weld. The average elastic stresses were compared to initial yield and to plastic instability conditions to modify conventional design formulas is presented. The factor is a linear function of the thicknesses of the parent plates attached by the fillet weld.

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).

Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework also relies extensively on accurate computation of harmonic forms. In this work we study the convergence properties of adaptive finite element methods (AFEM) for computing harmonic forms. We show that a properly defined AFEM is contractive and achieves optimal convergence rate beginning from any initial conforming mesh. This result is contrasted with related AFEM convergence results for elliptic eigenvalue problems, where the initial mesh must be sufficiently fine in order for AFEM to achieve any provable convergence rate.

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, Morley-Zienkiewicz element, modified-Zienkiewicz element et. al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.

We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model. Two different data assimilation situations with different stability properties are considered to illustrate the framework. Full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in appendix.

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers reliable and efficient a posteriori error estimators. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posed ness of the problem. Subsequently, applications of $C^0$ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings. Finally, we also discuss the variational discontinuous discretization method (without discretizing the control) and its corresponding error estimates.

We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is then used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.

In this paper we present a finite element method (FEM) for two-phase incompressible flows with moving contact lines. We use a sharp interface Navier-Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact we consider a class of continuum models which contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically the level-set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space (XFEM). We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. The numerical methods are first validated for relatively simple test problems, namely a stationary spherical droplet in contact with a flat wall and a spherical droplet on a flat wall that spreads or contracts to a stationary form. A further validation is done for a two-phase Couette flow with contact lines. To illustrate the robustness of our FEM we also present results of simulations for a problem with a curved contact wall and for a problem with more complicated contact line dynamics.

Discrete Mechanics based on finite element methods is presented in this paper. We also explore the relationship between this discrete mechanics and Veselov discrete mechanics. High order discretizations are constructed in terms of high order interpolations.

Discrete Mechanics based on finite element methods is presented in this paper. We also explore the relationship between this discrete mechanics and Veselov discrete mechanics. High order discretizations are constructed in terms of high order interpolations.

Lecture by Prof S K Lakshman Rao - on Finite Element Methods Note on the Shared Papers The papers shared here were authored by my late uncle, a respected Professor in his field. These documents are being shared solely for educational curiosity and reference. Please note that they reflect his personal views and interpretations on the subject matter at the time, and are intended as lecture notes only. They should not be considered as formal publications or definitive academic sources.

Lecture by Prof S K Lakshman Rao - on Finite Element Methods Note on the Shared Papers The papers shared here were authored by my late uncle, a respected Professor in his field. These documents are being shared solely for educational curiosity and reference. Please note that they reflect his personal views and interpretations on the subject matter at the time, and are intended as lecture notes only. They should not be considered as formal publications or definitive academic sources.