Showing papers on "Extended finite element method published in 1996"

Computational Differential Equations

Abstract: From the Publisher: This two volume introduction to the computational solution of differential equations uses a unified approach organized around the adaptive finite element method It presents a synthesis of mathematical modeling, analysis, and computation

Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems

Abstract: Ordinary Differential Equations.- The Analytical Behaviour of Solutions.- Numerical Methods for Second-Order Boundary Value Problems.- Parabolic Initial-Boundary Value Problems in One Space Dimension.- Analytical Behaviour of Solutions.- Finite Difference Methods.- Finite Element Methods.- Two Adaptive Methods.- Elliptic and Parabolic Problems in Several Space Dimensions.- Analytical Behaviour of Solutions.- Finite Difference Methods.- Finite Element Methods.- Time-Dependent Problems.- The Incompressible Navier-Stokes Equations.- Existence and Uniqueness Results.- Upwind Finite Element Method.- Higher-Order Methods of Streamline Diffusion Type.- Local Projection Stabilization for Equal-Order Interpolation.- Local Projection Method for Inf-Sup Stable Elements.- Mass Conservation for Coupled Flow-Transport Problems.- Adaptive Error Control.

A finite element method for interface problems in domains with smooth boundaries and interfaces

Abstract: This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.

Hierarchical bases and the finite element method

Abstract: In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the formulation of iterative methods for solving the large sparse sets of linear equations arising from finite element discretization.

Analysis of piezoelastic structures with laminated piezoelectric triangle shell elements

Abstract: In the recent development of active structural systems and microelectromechanical systems, piezoelectrics are widely used as sensors and actuators. Because of the limitations of theoretical and experimental models in design applications, finite element development and analysis are proposed and presented in this paper. A new laminated quadratic C° piezoelastic triangular shell finite element is developed using the layerwise constant shear angle theory. Element and system equations are also derived. The developed piezoelastic triangular shell element is used to model 1) a piezoelectric bimorph pointer and 2) a semicircular ring shell. Finite element (triangular shell finite element) solutions are compared closely with the theoretical, experimental, and finite element (thin solid finite element) results in the bimorph pointer case. Natural frequencies and distributed control effects of the ring shell with piezoelectric actuators of various length are also studied. Finite element analyses suggested that the inherent piezoelectric effect has little effect on natural frequencies of the ring shell. Vibration control effect increases as the actuator length increases, and it starts leveling off at the seven-patch (70%) actuator. Coupling and control spillover of lower natural modes are also observed.

Free mesh method: A new meshless finite element method

Abstract: A new meshless finite element method, named as the Free Mesh Method, is proposed in this paper. Once nodes are arranged in the domain to be analyzed, some temporary triangular elements are set around a node, i.e. a current central node. The contributions from the element matrices of the above temporary elements are assemebled to the total stiffness matrix. The above processes are performed on all the nodes in the domain. Finally, the solution is obtained by solving the total stiffness equation system as the usual finite element method. To demonstrate the effectiveness of the method, a simple two-dimensional heat conduction problem is solved.

Two-level additive Schwarz preconditioners for nonconforming finite element methods

Abstract: Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap.

Structural dynamic analysis by a time-discontinuous galerkin finite element method

Abstract: This paper studies a time-discontinuous Galerkin finite element method for structural dynamic problems, by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels A new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations By using the jumps of the displacements and the velocities in the total energy norm as error indicators, an adaptive time-stepping procedure for selecting the proper time step size is described Numerical examples including both single-DOF and multi-DOF problems are used to illustrate the performance of these algorithms Comparisons with the exact results and/or the results by the Newmark integration scheme are given It is shown that the time-discontinuous Galerkin finite element method discussed in this study possesses good accuracy (third order) and stability properties, its numerical implementation is not difficult, and the higher computational cost needed in each time step is compensated by use of a larger time step size

Adaptive finite-element method for electronic-structure calculations.

Abstract: We present a method which combines the finite-element method with the adaptive curvilinear coordinates, and the method is applied to the electronic-structure calculation of a model potential system. Comparison with other real-space methods, such as the finite-difference method is also made, and the efficiency of our method is examined. In addition, it has some desirable properties, and will be useful as a part of the O(N) method for self-consistent calculations. \textcopyright{} 1996 The American Physical Society.

Perfectly Matched Layers in the Discretized Space: An Analysis and Optimization

Abstract: The perfectly matched layer (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. Recently, it has been pointed out that this absorbing boundary condition is the same as coordinate stretching in the complex space. In this paper, the corresponding coordinate stretching is analyzed in the discretized space of Maxwell's equations as described by the Yee algorithm. The corresponding dispersion relationship is derived for a PML medium and then the problem of reflection from a single interface is solved. A perfectly matched interface is shown not to exist in the discretized space, even though it exists in the continuum space. Numerical simulations both using finite difference method and finite element method confirm that such discretization error exists. A numerical scheme using the finite element method is then developed to optimize the PML with respect to its parameters. Examples are given to demonstrate the performance of the optim...

Finite element model update via Bayesian estimation and minimization of dynamic residuals

Abstract: An algorithm is presented for updating finite element models based upon a minimization of dynamic residuals. The dynamic residual of interest is the force unbalance in the homogeneous form of the equations of motion arising from errors in the model`s mass and stiffness when evaluated with the identified modal parameters. The present algorithm is a modification and extension of a previously-developed Sensitivity-Based Element-By-Element (SB-EBE) method for damage detection and finite element model up- dating. In the present algorithm, SB-EBE has been generalized to minimize a dynamic displacement residual quantity, which is shown to improve test- analysis mode correspondence. Furthermore, the algorithm has been modified to include Bayesian estimation concepts, and the underlying nonlinear optimization problem has been consistently linearized to improve the convergence properties. The resulting algorithm is demonstrated via numerical and experimental examples to be an efficient and robust method for both localizing model errors and estimating physical parameters.

A model for analysis of arbitrary composite and porous microstructures with voronoi cell finite elements

Abstract: The Voronoi Cell Finite Element Model (VCFEM) has been successfully developed for materials with arbitrary microstructural distribution. In this method, the finite element mesh evolves naturally by Dirichlet Tessellation of the microstructure. Composite VCFEM for small deformation plasticity has been developed by expressing the element stresses in terms of polynomial expansions of location co-ordinates. Though this works well for discrete composites with inclusions, its effectiveness diminishes sharply for porous materials with voids. The effect worsens sharply with voids of arbitrary shapes. To overcome this limitation, a new way of defining stress functions is introduced in this paper. Based on a transformation method similar to the Schwarz–Christoffel conformal mapping, it introduces reciprocal stress functions that are derived to incorporate shape effects. Several numerical experiments are conducted to establish the strength of this formulation. The effect of various microstructural morphologies on the overall response and local variables are studied.

Finite element solution of nonlinear intrinsic equations for curved composite beams

Abstract: A geometrically nonlinear finite element analysis, based on a weak form of the geometrically exact intrisic equilibrium and constitutive equations, is presented for initially curved and twisted composite beams. Results for both nonlinear static deformation and linearized free vibration about the static state of deformation are obtained and compared with published exact and theoretical analyses in the literature for initially curved isotropic beams and for isotropic and composite beams with swept tips. T h e results agree very well with experiment.

Optimization strategies in unstructured mesh generation

Abstract: SUMMARY We propose a new optimization strategy for unstructured meshes that, when coupled with existing automatic generators, produces meshes of high quality for arbitrary domains in 3-D. Our optimizer is based upon a non-differentiable definition of the quality of the mesh which is natural for finite element or finite volume users: the quality of the worst element in the mesh. The dimension of the optimization space is made tractable by restricting, at each iteration, to a suitable neighbourhood of the worst element. Both geometrical (node repositioning) and topological (reconnection) operations are performed. It turns out that the repositioning method is advantageous with respect to both the usual node-by-node techniques and the more recent differentiable optimization methods. Several examples are included that illustrate the efficiency of the optimizer.

A finite element model for sound transmission through foam‐lined double‐panel structures

Abstract: In this paper, methods for coupling both elastic porous material (i.e., foam) and structural finite elements with either modal or finite element representations of acoustical system are presented. In addition, interface conditions are described for coupling elastic porous material finite elements with acoustical and structural finite elements in various configurations. The foam finite element is based on the elastic porous material theory of Biot. By considering sound transmission through layered systems placed in a waveguide, the accuracy of the coupled acoustical‐structural‐foam finite element model has been verified by comparing its transmission loss predictions with analytical solutions for the matching cases of infinite lateral extent. The constraint conditions at the edges of both the foam lining and the facing panels were found to have a significant effect on the normal incidence sound transmission loss of the double panel system at low frequencies.

Finite element analysis of orthogonal metal cutting mechanics

Abstract: The plane-strain finite element method is developed and applied to analyze the orthogonal metal cutting with continuous chip formation. Detailed work-material modeling, which includes the coupling of large strain, high strain-rate and temperature effects, is implemented. The versatility of the finite element method is demonstrated by presenting simulation results to complement the experimental measurements and to gain better understanding of the mechanics of the tool-chip contact and work-material deformation. The contour plots are used to show the distribution of parameters in the deformation zones. The finite difference method is applied to estimate the rate of change of parameters with respect to time. The Eulerian description of the deformation of work-material is also presented to show the variation at seven selected elements, which are expected to pass the deformation zones or go underneath the worn cutting tool.

Mixed finite element methods for flow in porous media

Abstract: Mixed nite element discretizations for problems arising in ow in porous medium applications are considered. We rst study second order elliptic equations which model single phase ow. We consider the recently introduced expanded mixed method. Combined with global mapping techniques, the method is suitable for full conductivity tensors and general geometry domains. In the case of the lowest order Raviart-Thomas spaces, quadrature rules reduce the method to cell-centered nite diierences, making it very eecient computationally. We consider problems with discontinuous coeecients on multiblock domains. To obtain accurate approximations, we enhance the scheme by introducing Lagrange multiplier pressures along subdomain boundaries and coeecient discontinuities. This modiication comes at no extra computational cost, if the method is implemented in parallel, using non-overlapping domain decomposition algorithms. Moreover, for regular solutions, it provides optimal convergence and discrete superconvergence for both pressure and velocity. We next consider the standard mixed nite element method on non-matching grids. We introduce mortar pressures along the non-matching interfaces. The mortar space is chosen to have higher approximability than the normal trace of the velocity spaces. The method is shown to be optimally convergent for all variables. Superconvergence for the subdomain pressures and, if the tensor coeecient is diagonal , for the velocities and the mortar pressures is also proven. We also consider the expanded mixed method on general geometry multiblock domains with non-matching grids. We analyze the resulting nite diierence scheme and show superconvergence for all variables. EEciency is not sacriiced by adding the mortar pressures. The computational complexity is shown to be comparable to the one on matching grids. Numerical results are presented, that verify the theory. iii We nally consider mixed nite element discretizations for the nonlinear multi-phase ow system. The system is reformulated as a pressure and a saturation equation. The methods described above are directly applied to the elliptic or parabolic pressure equation. We present an analysis of a mixed method on non-matching grids for the saturation equation of degenerate parabolic type. Acknowledgments To my wife Maria I would like to express my deep thanks to my advisor Mary Wheeler for her guidance and support. She has been both a mentor and a friend to me throughout my years at Rice. I also thank my former advisor Raytcho Lazarov, who is responsible for a great part of what I have achieved in my life. I am especially in debt to Todd Arbogast, the collaboration with …

The s-version of finite element method for laminated composites

Abstract: The s-version of the finite element method is developed for laminated plates and shells. By this technique the global domain is idealized using 2-D Equivalent Single Layer model. The regions where ESL model errs badly in capturing localized phenomena are superimposed by a stack of 3-D elements. Assumed strain formulation and selective polynomial order escalation in the two models, as well as fast iterative procedures, are employed to maintain a high level of computational efficiency.

Surgery Simulation Using fast Finite Elements

Abstract: This paper describes our recent work on real-time Surgery Simulation using Fast Finite Element models of linear elasticity [1]. In addition we discuss various improvements in terms of speed and realism.

Finite element micromechanical modeling of the cochlea in three dimensions

Abstract: A new cochlear modeling technique has been developed in which the number of assumptions required in model formulation is significantly less than in previous modeling studies. The main new feature of the method is that it allows individual cellular and membrane components of the organ of Corti to be embedded within the model fluid in their true structural positions, with connections to neighboring elements reflecting anatomical geometry. The cochlea is divided into a three-dimensional finite element (3-D FE) network of nodes, connected by branches representing the local mechanical properties. The model system of simultaneous equations, obtained by applying continuity at each node, is solved iteratively using a variant of the conjugate gradient method. Here the formulation and implementation of the 3-D FE method are described. Force generation by outer hair cells is included and results are presented which demonstrate the effect of tectorial membrane and Deiters' cell mechanical properties on the effectiveness of the cochlear amplifier.

A two-level additive Schwarz preconditioner for nonconforming plate elements

Abstract: A two-level additive Schwarz preconditioner is developed for the systems resulting from the discretizations of the plate bending problem by the Morley finite element, the Fraeijs de Veubeke finite element, the Zienkiewicz finite element and the Adini finite element. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of a generous overlap.