On the hybrid-mixed formulation of C0 curved beam elements

Two C⁰ curved beam elements based on the hybrid-mixed formulation are studied in the form of membrane-shear locking, mesh convergence, and stress predictions. At the element level, both the displacement and stress fields are approximated separately. The stress parameters are then eliminated from the stationary condition of the Hellinger-Reissner variational principle so that the standard stiffness equations are obtained. The stress functions are chosen from two important considerations: (i) kinematic deformation modes must be avoided, and (ii) the constraint index counting of the element, when applied to limiting cases, must be equal to or greater than one. Based on these considerations, two curved beam elements are derived by including the effect of shear deformation and with linear and quadratic displacement fields. The elements are found to be lock-free for thin-walled beams. Several numerical examples are given to demonstrate the performance of the two curved elements.     

Implementation of an hybrid-mixed stress model based on the use of wavelets

This paper illustrates the use of wavelets defined on the interval as approximation functions in hybrid-mixed stress finite element models applied to the solution of linear elastic plate stretching and plate bending problems. Special attention is given to the algorithms and techniques involved in the generation and manipulation of such functions. A set of numerical examples is presented to illustrate the use of the hybrid-mixed model.     

Free-vibration analysis of arches based on the hybrid-mixed formulation with consistent quadratic stress functions

The well-known two-node hybrid-mixed element with linear displacement interpolation functions and constant stress resultant functions proposed by Saleeb yields locking-free results in static analysis but shows unsatisfactory results in vibration analysis of arches. In this study, we investigate the role of higher-order interpolation functions and consistent stress resultant functions in developing an effective two-node hybrid-mixed finite element for free-vibration analysis of arches with a rectangular cross-section. The present thick element considering shear deformation is based on the Hellinger–Reissner variational principle and introduces additional nodeless degrees of freedom for displacement field interpolation in order to enhance substantially the numerical accuracy especially in predicting high vibration modes. In the performance evaluation of the present hybrid-mixed element, we examine the effect of the nodeless internal displacement functions, the field-consistent stress resultant parameters, the approximation of compliance matrix, and the shear correction factor in curved beam vibrations. Several numerical examples confirm the validity of the present element and also illustrate the dynamic characteristics of arches depending on the curvature, aspect ratio and boundary conditions, etc.     

Highly scalable parallel algorithms for sparse matrix factorization

In this paper, we describe scalable parallel algorithms for symmetric sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1,024 processors on a Gray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algorithms substantially improve the state of the art in parallel direct solution of sparse linear systems-both in terms of scalability and overall performance. It is a well known fact that dense matrix factorization scales well and can be implemented efficiently on parallel computers. In this paper, we present the first algorithms to factor a wide class of sparse matrices (including those arising from two- and three-dimensional finite element problems) that are asymptotically as scalable as dense matrix factorization algorithms on a variety of parallel architectures. Our algorithms incur less communication overhead and are more scalable than any previously known parallel formulation of sparse matrix factorization. Although, in this paper, we discuss Cholesky factorization of symmetric positive definite matrices, the algorithms can be adapted for solving sparse linear least squares problems and for Gaussian elimination of diagonally dominant matrices that are almost symmetric in structure. An implementation of one of our sparse Cholesky factorization algorithms delivers up to 20 GFlops on a Gray T3D for medium-size structural engineering and linear programming problems. To the best of our knowledge, this is the highest performance ever obtained for sparse Cholesky factorization on any supercomputer     

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.     

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.     

On some simplified models for optimal design of structural systems

Some simplified models for optimal design of large structural systems are reviewed. The design variables represent the members' cross-sections and the constraints are related to stress, displacement and design considerations. Using displacement analysis formulation, several explicit behavior models which do not involve multiple implicit analyses are presented. It is shown that approximations of the displacements, often used in optimization of large systems, essentially lead to solutions which do not satisfy equilibrium.Approximations along a line in the design variables space are developed, and simplifications based on the virtual load method and inverse design variables formulations are presented. The relationships between the various models are derived; it is shown that, under the usual assumption of linear relationship between the stiffness matrix elements and the design variables, some of the approximations become equivalent.     

Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.     

Weak Galerkin method for the Biot’s consolidation model

We develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.     

An element-based displacement preconditioner for linear elasticity problems

Finite element analysis of problems in structural and geotechnical engineering results in linear systems where the unknowns are displacements and rotations at nodes. Although the solution of these systems can be carried out using either direct or iterative methods, in practice the matrices involved are usually very large and sparse (particularly for 3D problems) so an iterative approach is often advantageous in terms of both computational time and memory requirements. This memory saving can be further enhanced if the method used does not require assembly of the full coefficient matrix during the solution procedure. One disadvantage of iterative methods is the need to apply preconditioning to improve convergence. In this paper, we review a range of established element-based preconditioning methods for linear elastic problems and compare their performance with a new method based on preconditioning with element displacement components. This new method appears to offer a significant improvement in performance.     

Mixed stabilized finite element methods in nonlinear solid mechanics: Part I: Formulation

This paper exploits the concept of stabilized finite element methods to formulate stable mixed stress/displacement and strain/displacement finite elements for the solution of nonlinear solid mechanics problems. The different assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation.     

p-Version axisymmetric shell element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order p_ξ and p_η. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to p_ξ + 1 and p_η+ 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C⁰ continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element.

The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete axisymmetric state of stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells. The formulation presented here removes virtually all of the drawbacks present in the existing GNL axisymmetric shell finite element formulations and has many additional benefits. First, the currently available GNL axisymmetric shell finite element formulations are based on fixed interpolation order and thus are not hierarchical and have no mechanism for p-level change. Secondly, the element displacement approximations in the existing formulations are either based on linearized (with respect to nodal rotation) displacement fields in which case a true Lagrangian formulation is not possible and the load step size is severely limited or are based on nonlinear nodal rotation functions approach in which case though the kinematics of deformation is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. Such limitations and difficulties do not exist in the present formulation. The hierarchical displacement approximation used here does not involve traditional nodal rotations that have been used in the existing shell element formulations, thus the difficulties associated with their use are not present in this formulation.

Incremental equations of equilibrium are derived and solved using the standard Newton method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the preset formulation. The results obtained from the present formulation are compared with those available in the literature.     

Application of radial basis functions to linear and nonlinear structural analysis problems

The basic characteristic of the techniques generally known as meshless methods is the attempt to reduce or even to eliminate the need for a discretization (at least, not in the way normally associated with traditional finite element techniques) in the context of numerical solutions for boundary and/or initial value problems.The interest in meshless methods is relatively new and this is why, despite the existence of various applications of meshless techniques to several problems of mechanics (as well as to other fields), these techniques are still relatively unknown to engineers. Furthermore, and compared to traditional finite elements, it may be difficult to understand the physical meaning of the variables involved in the formulations.As an attempt to clarify some aspects of the meshless techniques, and simultaneously to highlight the ease of use and the ease of implementation of the algorithms, applications are made, in this work, to structural analysis problems.The technique used here consists of the definition of a global approximation for a given variable of interest (in this case, components of the displacement field) by means of a superposition of a set of conveniently placed (in the domain and on the boundary) radial basis functions (RBFs).In this work various types of one-dimensional problems are analyzed, ranging from the static linear elastic case, free vibration and linear stability analysis (for a beam on elastic foundation), to physically nonlinear (damage models) problems. To further complement the range of problems analysed, the static analysis of a plate on elastic foundation was also addressed. Several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global RBFs. The results obtained show that RBF collocation leads to good approximations and very high convergence rates.     

Comparison of different solution algorithms for collocated method of MCIM to calculate steady and unsteady incompressible flows on unstructured grids

In this paper, the collocated method of MCIM (Mass Corrected Interpolation Method) is employed for solving two-dimensional incompressible flows on unstructured grid systems. A control-volume-based finite element (CVFEM) approach has been taken in which conservation equations are formed for each control volume throughout the solution domain. In this study, three different algorithms are proposed and investigated. In the first algorithm a fully coupled formulation is used, however in the second algorithm a segregated approach without pressure correction is employed. In the third solution algorithm, the scheme of MCIM is implemented to SIMPLE-like segregated algorithm. The linear system of equations obtained in these algorithms is solved using a direct sparse solver. The accuracy and performance of the proposed algorithms are demonstrated by solving different steady and unsteady benchmark problems.     

Sparse coding via thresholding and local competition in neural circuits

While evidence indicates that neural systems may be employing sparse approximations to represent sensed stimuli, the mechanisms underlying this ability are not understood. We describe a locally competitive algorithm (LCA) that solves a collection of sparse coding principles minimizing a weighted combination of mean-squared error and a coefficient cost function. LCAs are designed to be implemented in a dynamical system composed of many neuron-like elements operating in parallel. These algorithms use thresholding functions to induce local (usually one-way) inhibitory competitions between nodes to produce sparse representations. LCAs produce coefficients with sparsity levels comparable to the most popular centralized sparse coding algorithms while being readily suited for neural implementation. Additionally, LCA coefficients for video sequences demonstrate inertial properties that are both qualitatively and quantitatively more regular (i.e., smoother and more predictable) than the coefficients produced by greedy algorithms.     

Stability and phase speed for various finite element formulations of the advection equation

This paper analyzes the stability and accuracy of various finite element approximations to the linearized two-dimensional advection equation. Four triangular elements with linear basis functions are included along with a rectangular element with bilinear basis functions. In addition, second-and fourth-order finite difference schemes are examined for comparison. Time is discretized with the leapfrog method. The criss-cross triangle formulation is found to be unstable. The best schemes are the isosceles triangles with linear functions and the rectangles with bilinear basis functions.     

Stabilized finite element formulation for elastic–plastic finite deformations

This paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic–plastic solids, such as metals. An updated Lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation. This formulation circumvents the restriction on the displacement and pressure fields due to the Babuška–Brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit, typical in metal forming applications. Moreover, it facilitates the use of low order simplex elements (i.e. P1/P1), reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary. Linearization of the weak form is derived for implementation into a finite element code. Numerical experiments with P1/P1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis.     

Topology optimization of nonlinear structures with damping under arbitrary dynamic loading

This study presents an extended unit load method in which the displacement of a chosen degree of freedom (DOF) in a nonlinear structure under arbitrary dynamic loading is expressed as an integration of mutual strain energy density over a continuum domain. This new integral formulation for the displacement of a chosen DOF is developed by using the virtual work principle and can be used for linear or nonlinear structural behaviours. The integral form of the displacement is then used to develop new formulations for structural topology optimization involving arbitrary dynamic loading using the moving iso-surface threshold (MIST) method. Presented are two specific topology optimization problems with two objective functions: (a) to minimize the peak of a chosen displacement; or (b) to minimize the average power spectral density (PSD) of the chosen displacement over a finite time interval. New MIST formulations and algorithms are developed for solving two damping topology optimization problems of a structure under arbitrary dynamic loading, with or without large displacements, and having cellular damping materials with multi-volume fractions. Several numerical examples are presented to demonstrate the validity and efficiency of the presented unit load method and the MIST formulations and algorithms.     

Interpolation of rotation and motion

In Cosserat solids such as shear deformable beams and shells, the displacement and rotation fields are independent. The finite element implementation of these structural components within the framework of flexible multibody dynamics requires the interpolation of rotation and motion fields. In general, the interpolation process does not preserve fundamental properties of the interpolated field. For instance, interpolation of an orthogonal rotation tensor does not yield an orthogonal tensor, and furthermore, does not preserve the tensorial nature of the rotation field. Consequently, many researchers have been reluctant to apply the classical interpolation tools used in finite element procedures to interpolate these fields. This paper presents a systematic study of interpolation algorithms for rotation and motion. All the algorithms presented here preserve the fundamental properties of the interpolated rotation and motion fields, and furthermore, preserve their tensorial nature. It is also shown that the interpolation of rotation and motion is as accurate as the interpolation of displacement, a widely accepted tool in the finite element method. The algorithms presented in this paper provide interpolation tools for rotation and motion that are accurate, easy to implement, and physically meaningful.     

Isotropic Energies, Filters and Splines for Vector Field Regularization

The aim of this paper is to propose new regularization and filtering techniques for dense and sparse vector fields, and to focus on their application to non-rigid registration. Indeed, most of the regularization energies used in non-rigid registration operate independently on each coordinate of the transformation. The only common exception is the linear elastic energy, which enables cross-effects between coordinates. Cross-effects are yet essential to give realistic deformations in the uniform parts of the image, where displacements are interpolated.In this paper, we propose to find isotropic quadratic differential forms operating on a vector field, using a known theorem on isotropic tensors, and we give results for differentials of order 1 and 2. The quadratic approximation induced by these energies yields a new class of vectorial filters, applied numerically in the Fourier domain. We also propose a class of separable isotropic filters generalizing Gaussian filtering to vector fields, which enables fast smoothing in the spatial domain. Then we deduce splines in the context of interpolation or approximation of sparse displacements. These splines generalize scalar Laplacian splines, such as thin-plate splines, to vector interpolation. Finally, we propose to solve the problem of approximating a dense and a sparse displacement field at the same time. This last formulation enables us to introduce sparse geometrical constraints in intensity based non-rigid registration algorithms, illustrated here on intersubject brain registration.