Quadratic Finite Element Research Articles

A projection method to extract biological membrane models from 3D material models

This paper presents a projection method for deriving membrane models from the corresponding three-dimensional material models. As a particular example the anisotropic Holzapfel–Gasser–Ogden model is considered. The projection procedure is based on the kinematical and constitutive assumptions of a general membrane theory, considering the membrane to be a general two-dimensional manifold. By assuming zero transverse stress, the Lagrange multiplier associated with the incompressibility constraint can be eliminated from the formulation. The resulting nonlinear model is discretized and linearized within the finite element method. Several numerical examples are shown, considering quadratic Lagrange and NURBS finite elements. These show that the proposed model is in very good agreement with analytical solutions and with full 3D finite element computations.

Weak form quadrature element method for accurate free vibration analysis of thin skew plates

In this paper, a novel weak form quadrature element method (QEM) is proposed for accurate free transverse vibration analysis of thin isotropic skew plates with general boundary conditions. In the formulation of the stiffness matrix, the first and second order derivatives of the shape functions at integration points are computed explicitly by using the differential quadrature rule. This leads to a great simplicity in formulating an N×N-node quadrature plate element and a large reduction of programming effort. Different from the existing weak form quadrature element method or differential quadrature finite element method, the element nodes can be either the same or different from the integration points. Convergence studies are performed. Free vibration of skew thin plates with various skew angles and different combinations of boundary conditions is analyzed. It is shown that although the assumed displacement field does not explicitly consider the bending moment singularities at the obtuse angles, the proposed QEM can yield accurate frequencies even for the thin isotropic skew plate with large skew angles.

A new approach on numerical solutions of the Improved Boussinesq type equation using quadratic B-spline Galerkin finite element method

In the present manuscript, some numerical solutions of an Improved Boussinesq type equation are obtained by means of quadratic B-spline Galerkin finite element method. Then, error norms L2 and L∞ have been calculated to test the accuracy of the current method. In the manuscript, solitary wave movement and interaction of solitary–antisolitary waves are considered as test problems.

Analysis of composite plates using a layerwise theory and a differential quadrature finite element method

A layerwise shear deformation theory for composite laminated plates is discretized using a differential quadrature finite element method (DQFEM). The DQFEM is a weak-form differential quadrature method that can provide highly accurate results using only a few sampling points. The layerwise theory proposed by Ferreira is based on an expansion of Mindlin’s first-order shear deformation theory in each layer and results for a laminated plate with three layers were presented as example in the original paper. This work generalized the layerwise theory to plates with any number of layers. The combination of the DQFEM with Ferreira’s layerwise theory allows a very accurate prediction of the field variables. Laminated composite and sandwich plates were analyzed. The DQFEM solutions were compared with various models in literature and especially showed very good agreements with the exact solutions in literature that was based on a similar layerwise theory. The analysis of composite plates based on Ferreira’s layerwise theory indicates that the DQFEM is an effective method for high accuracy analysis of large-scale problems.

Deformation simulation using cubic finite elements and efficient p-multigrid methods

We present a novel p-multigrid method for efficient simulation of corotational elasticity with higher-order finite elements. In contrast to other multigrid methods proposed for volumetric deformation, the resolution hierarchy is realized by varying polynomial degrees on a tetrahedral mesh. The multigrid approach can be either used as a direct method or as a preconditioner for a conjugate gradient algorithm. We demonstrate the efficiency of our approach and compare it to commonly used direct sparse solvers and preconditioned conjugate gradient methods. As the polynomial representation is defined w.r.t. the same mesh, the update of the matrix hierarchy necessary for corotational elasticity can be computed efficiently. We introduce the use of cubic finite elements for volumetric deformation and investigate different combinations of polynomial degrees for the hierarchy. We analyze the applicability of cubic finite elements for deformation simulation by comparing analytical results in a static and dynamic scenario and demonstrate our algorithm in dynamic simulations with quadratic and cubic elements. Applying our method to quadratic and cubic finite elements results in a speed-up of up to a factor of 7 for solving the linear system.

A quadratic temporal finite element method for linear elastic structural dynamics

Recently, in order to overcome the key shortcoming in Hamilton’s principle, the extended framework of Hamilton’s principle was formulated. This new variational framework provides a sound base to develop novel temporal finite element methods with the proper use of initial conditions to the strong form. As its initial and practical application in structural dynamic analysis, this paper presents a quadratic temporal finite element method for linear elastic multi-degrees-of-freedom systems. Numerical features of the developed method are analytically and numerically investigated with comparison to the currently dominant method; the results highlight the improved performance of the proposed method.

An inner–outer nonlinear programming approach for constrained quadratic matrix model updating

The Quadratic Finite Element Model Updating Problem (QFEMUP) concerns with updating a symmetric second-order finite element model so that it remains symmetric and the updated model reproduces a given set of desired eigenvalues and eigenvectors by replacing the corresponding ones from the original model. Taking advantage of the special structure of the constraint set, it is first shown that the QFEMUP can be formulated as a suitable constrained nonlinear programming problem. Using this formulation, a method based on successive optimizations is then proposed and analyzed. To avoid that spurious modes (eigenvectors) appear in the frequency range of interest (eigenvalues) after the model has been updated, additional constraints based on a quadratic Rayleigh quotient are dynamically included in the constraint set. A distinct practical feature of the proposed method is that it can be implemented by computing only a few eigenvalues and eigenvectors of the associated quadratic matrix pencil.

An approach to time fractional gas dynamics equation: Quadratic B-spline Galerkin method

In the present article, a quadratic B-spline finite element Galerkin method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the L1 discretization formula is applied to the equation in time. A numerical example is given and the obtained results show the accuracy and efficiency of the method. Therefore, the present method can be used as an efficient alternative one to find out the numerical solutions of other both linear and nonlinear fractional differential equations available in the literature.

Linear differential operators on bivariate spline spaces and spline vector fields

We consider the application of standard differentiation operators to spline spaces and spline vector fields defined on triangulations in the plane. In particular, we explore the use of Bernstein–Bezier techniques for answering questions such as: What are the images or the kernels, and their dimensions, of partial derivative, gradient, divergence, curl, or Laplace, operators. We also describe a particular continuous piecewise quadratic finite element whose nodal parameters are function and divergence (or curl) values.

A simple finite element method for boundary value problems with a Riemann–Liouville derivative

We consider a boundary value problem involving a Riemann–Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and L2(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm–Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.

A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem

Abstract A residual based a posteriori error estimator is derived for a quadratic finite element method (FEM) for the elliptic obstacle problem. The error estimator involves various residuals consisting of the data of the problem, discrete solution and a Lagrange multiplier related to the obstacle constraint. The choice of the discrete Lagrange multiplier yields an error estimator that is comparable with the error estimator in the case of linear FEM. Further, an a priori error estimate is derived to show that the discrete Lagrange multiplier converges at the same rate as that of the discrete solution of the obstacle problem. The numerical experiments of adaptive FEM show optimal order convergence. This demonstrates that the quadratic FEM for obstacle problem exhibits optimal performance.

A stabilized single-point finite element formulation for three-dimensional dynamic analysis of saturated soils

A low-order single-point quadrature finite element suitable of dynamic 3D analysis of saturated soils is presented. The new element uses a u-p formulation to consider the interaction of the pore fluid and solid skeleton. An hourglass stabilization scheme facilitates single-point integration for the solid phase terms, and non-residual pressure field stabilization is used to facilitate equal-order interpolation for the two phases and improve element behavior in the incompressible–impermeable limit. Several numerical examples are presented to verify the new element formulation and demonstrate its response in a geotechnical application.

Three-dimensional J-integral evaluation for finite strain elastic–plastic solid using the quadratic tetrahedral finite element and automatic meshing methodology

In this paper, the three-dimensional J-integral for finite deformation problems is critically concerned when it is computed by the domain integral method for the quadratic tetrahedral finite element. When a cracked body undergoes a large elastic–plastic deformation, the domain independent property is not guaranteed. Evaluation procedures that guarantee the domain independent property by including the terms involving the spatial derivatives of the strain energy density, etc., are proposed. The proposed methodology includes a way to compute such derivatives by adopting the concept of superconvergent patch recovery method. Finally, numerical examples demonstrate that proposed procedures can guarantee the domain independent property.

Optimal Convergence for Discrete Variational Inequalities Modelling Signorini Contact in 2D and 3D without Additional Assumptions on the Unknown Contact Set

The basic $H^1$-finite element error estimate of order $h$ with only $H^2$-regularity on the solution has not been yet established for the simplest two-dimensional Signorini problem approximated by a discrete variational inequality (or the equivalent mixed method) and linear finite elements. To obtain an optimal error bound in this basic case and also when considering more general cases (e.g., the three-dimensional problem, quadratic finite elements), additional assumptions on the exact solution (in particular on the unknown contact set, see [Z. Belhachmi and F. Ben Belgacem, Math. Comp., 72 (2003), pp. 83--104; S. Hueber and B. Wohlmuth, SIAM J. Numer. Anal., 43 (2005), pp. 156--173; B. Wohlmuth, A. Popp, M. Gee, and W. Wall, Comput. Mech. 49 (2012), pp. 735--747] had to be used. In this paper, we consider finite element approximations of the two-dimensional and three-dimensional Signorini problems with linear and quadratic finite elements. In the analysis, we remove all the additional assumptions and pr...

Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element

In this article, using the well-known Superconvergent Patch Recovery (SPR) method, we present a gradient superconvergence post-processing scheme for the tensor-product quadratic pentahedral finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over fully uniform meshes. The supercloseness property of the gradients between the finite element solution $u_h$ and the tensor-product quadratic interpolation $\Pi u$ is first given. Then we show that the gradient recovered from the finite element solution by using the SPR method is superconvergent to $\nabla u$ at interior vertices.

Continuous Finite Element Methods of Molecular Dynamics Simulations

Molecular dynamics simulations are necessary to perform very long integration times. In this paper, we discuss continuous finite element methods for molecular dynamics simulation problems. Our numerical results aboutABdiatomic molecular system andA2Btriatomic molecules show that linear finite element and quadratic finite element methods can better preserve the motion characteristics of molecular dynamics, that is, properties of energy conservation and long-term stability. So finite element method is also a reliable method to simulate long-time classical trajectory of molecular systems.

A global–local theory with stress recovery and a new post-processing technique for stress analysis of asymmetric orthotropic sandwich plates with single/dual cores

While the 3D elasticity and the available local and zigzag theories encounter inaccuracies for very thin plates, results of the global theories become unreliable for thick plates with severe transverse variations in the material properties. In the present research, a quadratic finite element global–local sandwich plate theory with elasticity correction (stress recovery) is proposed for static stress and displacement analysis of sandwich plates with orthotropic face sheets and single or dual cores. Since the transverse shear stresses are derived based on the three-dimensional theory of elasticity, the continuity condition of the transverse shear stresses is satisfied at the interfaces between the layers, a priori. The presented theory not only leads to higher accuracies in comparison to the available high-order zigzag theories in some cases (especially, for soft or dual cores) but also it is computationally more economic. Results show that the results of the available zigzag and global–local theories may encounter inaccuracy problems not only for huge numbers of the sub-layers, but also for small numbers of the orthotropic layers.

A generalized multiscale finite element method for the Brinkman equation

In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model in two dimensions. In the fine grid, we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space. The stability of the mixed GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical examples are presented to illustrate the effectiveness of the algorithm.

A coupled finite–hierarchic infinite element method for an inhomogeneous transversely isotropic soil layer under irregularly distributed strip load

In this paper, a coupled finite–hierarchic infinite element procedure is presented for an inhomogeneous transversely isotropic soil layer under irregularly distributed strip load. Elasticity and shear moduli follow a power law variation with depth. Quadratic quadrangular finite elements are coupled with a coordinate ascent hierarchic infinite element along the bounded edge using a least square approximation. The element mesh is fixed. Convergence is reached by increasing the polynomial degree. Novel results are presented for the vertical displacement and normal stress induced by uniform, parabolic, and triangular strip loads. The method is found to converge rapidly when the polynomial degree is increased. The calculated results for a homogeneous isotropic soil layer under a triangular strip load are compared with exact solutions and show excellent agreement. A parametric analysis reveals that inhomogeneity, transverse isotropy, layer depth, and form of strip load have an important effect on the soil layer physical behavior.

Quadratic mixed finite element approximations of the Monge–Ampère equation in 2D

We give error estimates for a mixed finite element approximation of the two-dimensional elliptic Monge---Ampere equation with the unknowns approximated by Lagrange finite elements of degree two. The variables in the formulation are the scalar variable and the Hessian matrix.