Finite Element Stiffness Matrix Research Articles

Semi‐analytical integration of the 8‐node plane element stiffness matrix using symbolic computation

AbstractThe semi‐analytical integration of an 8‐node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super‐parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic‐manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi‐analytical integration and numerical integration were made. It was demonstrated that semi‐analytical integration required less CPU time than conventional numerical integration (using Gaussian‐Legendre quadrature) to obtain the stiffness matrix. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Vibration analysis of rotors for the identification of shaft misalignment Part 1: Theoretical analysis

Shaft misalignment in machinery causes preload forces to be generated in couplings which are then transmitted to the different machine components, thus reducing their lifetime. Shaft misalignment is a major cause of vibration in machines. Based on tests on coupling stiffness, a new coupling finite element stiffness matrix has been deduced. This has been used in the finite element analysis of a two-rotor system connected by a flexible coupling, to calculate the mechanical vibration resulting from mixed angular and parallel shaft misalignments with residual unbalance. The calculated vibration spectra were determined for two flexible couplings: a three-pin Renold coupling and a three-jaw Lovejoy coupling. The results and analysis indicate that the vibration generated by shaft misalignment is caused by the variation in coupling stiffness on rotation, and that the forcing frequencies generated are harmonics of the shaft's speed and directly depend on the frequency of the variation in coupling stiffness. In addition, the amplitudes of the measured vibratory components were found to rely directly upon the frequency response function that is related to the coupling and measurement points.

Optimal finite element mesh for elliptic equation of divergence form

We consider an optimal triangular mesh minimizing the condition number of the finite element stiffness matrix for an elliptic equation −∑ ∂ ∂x i (a ij ∂u ∂x j )=f , u| ∂Ω =g . Using a sharp bound for the condition number of the stiffness matrix, it is shown that the element of the optimal uniform triangular mesh is equilateral with respect to the metric which is the inverse of the coefficient matrix in the equation. It is verified by numerical examples that such a mesh is really effective in reducing the condition number of the stiffness matrix. In addition, we suggest an algorithm generating a mesh in which every element is almost equilateral with respect to a metric.

Effects of microscale material randomness on the attainment of optimal structural shapes

Classical procedures of shape optimization of engineering structures implicitly assume the existence of a hypothetical perfectly homogeneous continuum --- they do not recognize the presence of any microscale material randomness. By contrast, the present study investigates this aspect for the paradigm of a Michell truss with minimum compliance (maximum stiffness) that has a prescribed weight. The problem involves a stochastic generalization of the topology optimization method implemented in the commercial Altair's OptiStruct computer code. In particular, this generalization allows for the dependence of each finite element's stiffness matrix on the actual microstructure contained in the given element's domain. Contrary to intuition, stochastic material properties may improve the compliance of optimal design. This is because the optimization is performed on a given random distribution, so that the design process has an opportunity to choose `stiffer' cells and discard those with weaker material. The paper does not aim for a robust design process, but tries to answer a simpler intermediate question: how the random fluctuation of material properties influences a structure that has been designed using classical continuum-based optimization algorithms.

Monotone iterative methods for the adaptive finite element solution of semiconductor equations

Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive finite element solution of semiconductor equations in terms of the Slotboom variables. The adaptive meshes are generated by the 1-irregular mesh refinement scheme. Based on these unstructured meshes and a corresponding modification of the Scharfetter–Gummel discretization scheme, it is shown that the resulting finite element stiffness matrix is an M-matrix which together with the Shockley–Read–Hall model for the generation–recombination rate leads to an existence–uniqueness–comparison theorem with simple upper and lower solutions as initial iterates. Numerical results of simulations on a MOSFET device model are given to illustrate the accuracy and efficiency of the adaptive and monotone properties of the present methods.

Matrix Analysis of Shear Lag and Shear Deformation in Thin-Walled Box Beams

This paper presents an initial value solution of the static equilibrium differential equations of thin-walled box beams, considering both shear lag and shear deformation. This solution was used to establish the related finite element stiffness matrix and equivalent nodal forces vector. In the procedure a special shear-lag-induced bimoment is introduced, so that the analysis of shear lag and shear deformation of thin-walled box beams is admitted into the program system of the matrix-displacement method. The present procedure can be used to analyze accurately the shear lag and shear deformation effects for thin-walled box beams, especially for some complex structures (such as continuous box girders and box beams with varying cross section, etc.). The numerical results obtained by the present procedure are consistent with the results of model tests and predictions of the finite shell element method or finite difference approach.

Finite‐Element Waveform Inversion of Acoustic Wave Equation in the Frequency Domain

AbstractWe deduce the finite‐element acoustic wave equation in the frequency domain. In order to eliminate boundary reflection, the absorbing boundary conditions of Clayton‐Engquist paraxial wave equation are introduced to the frequency domain. The finite‐element stiffness matrix and mass matrix are compressed for storage. The solutions of forward modeling are obtained by using the generalized conjugate gradient algorithm. On these bases, we deduce the Jacobi matrix representing the relation between the wavefield data residuals δ Ů and the element material‐property adjusted values δ λ at a certain frequency. Using the differences δ Ů between the surface 2D shot recorded data and theoretical modelling data, the adjusted values δ λ can be obtained iteratively. Because of the limitation of the computer's storage, larger number of unknowns is not permitted. The measure of compressing and assembling the element Jacobi matrix coefficients in the same medium is also proposed. By using this method, the unknowns' number of inversion is reduced. Combined with the conjugate gradient algorithm, only a few frequencies in valid‐wave domain are needed in this inversion. Some numerical examples of the modeling and inversion are given. The effectivity of the method is proved using the results.

Trefftz Finite and Boundary Element Method

Part 1: finite element technique shape functions and element stiffness matrix brief historical background basic relationships in engineering problems modified variational principles the concept of T-complete solution comparison of T-elements with conventional finite elements comparison of T-elements with boundary elements. Part 2: potential problems - introduction statement of the problem T-complete functions assumed fields generation of element matrix equation rank condition special purpose functions sensitivity to mesh distortion orthotropic case the Helmholtz equation HT-element with boundary traction frame frameless T-elements. Part 3: linear elastostatics - introduction linear theory of elasticity assumed fields in plane elasticity T-complete functions variational formulations element stiffness equation special-purpose elements p-extension approach three-dimensional elasticity numerical examples. Part 4: thin plates - introduction thin plate theory assumed field T-complete functions and particular solutions variational formulations for plate bending generation of element stiffness matrix p-method elements special purpose functions Extension to thin plates on elastic foundation Two alternative plate bending p-elements Numerical examples and assessment. Chapter 5 - Thick Plates - Introduction Basic equations for Reissner-Mindlin plate theory Assumed fields and particular solution Variational formulation for HT thick plate elements Implementation of the new family of HT elements A 12 DOF quadrilateral element free of shear locking Extension to thick plates on elastic foundation Sensitivity to mesh distortion Numerical assessment. Chapter 6 - Transient Heat Conduction - Introduction Elements of heat conduction Time step formula Element matrix formulations T-complete functions and particular solutions Numerical examples. Chapter 7 - Geometrically Nonlinear Analysis of Plate Bending Problems - Introduction Basic equations of nonlinear thin plate bending Assumed fields and Trefftz functions Particular solutions Modified variational principle Element matrix Iterative scheme Extension to post-buckling thin plates on elastic foundation Geometrically nonlinear analysis of thick plates Numerical examples. Chapter 8 - Elastoplasticity - Introduction Time discretization Basic relations Assumed fields Constraints on the approximation functions Finite element equilibrium and compatibility equations Finite element equations Finite element governing system. Chapter 9 - Dynamics of Plate Bending Problems - Introduction Basic equations Time-stepping formulation Numerical examples. Chapter 10 - Trefftz Boundary Element Method - Introduction Potential problems Plane elasticity Thin plate bending Moderately thick plates.

Preconditioning C1 Lagrange polynomial spline collocation method of elliptic equations by finite element method

This paper studies a preconditioning polynomial spline collocation method for an elliptic differential equation by a finite element stiffness matrix. For which, we first show that the decay rate of the C 1 Lagrange polynomial spline increases with increasing degree and second show that the distribution of eigenvalues for the established preconditioning system is well bounded in terms of the number of unknowns for a fixed degree of polynomial spline.

Crack detection and vibration behavior of cracked beams

A theoretical and experimental dynamic behavior of different multi-beams systems containing a transverse crack is presented. The additional flexibility that the crack generates in its vicinity is evaluated using the strain energy density function given by the linear fracture mechanic theory. Based on this flexibility, a new cracked finite element stiffness matrix is deduced, which can be used subsequently in the FEM analysis of crack systems. The proposed element is used to evaluate the dynamic response of a cracked free–free beam and a U-frame when a harmonic force is applied. The resulting parametrically excited system is non-linear and the equations of motion are solved using the Hilbert, Hughes and Taylor integration method implemented using a Matlab software platform. Some useful conclusions for diagnosing cracked beams systems are proposed.

Unsymmetrically Loaded Cylindrical Tank on Elastic Foundation

The analysis of axisymmetric shells of arbitrary shapes subjected to arbitrary loading has attracted considerable attention. If one considers the Fourier series of the circumference direction of loads and displacements, an axisymmetric tank subjected to a nonaxisymmetric load can be treated as a 1D problem. This paper combines the foundation and element stiffness matrices to make a circular plate element resting on an elastic foundation. This finite-element stiffness matrix is then converted into a transfer matrix to reduce the unknown quantity of the degrees of freedom of the element, no matter how many elements the shell structure has. Nonaxisymmetrically loaded cylindrical tanks resting on an elastic foundation are analyzed by the transfer matrix method. The calculated results of the proposed method to the nonaxisymmetrically loaded cylindrical tanks on an elastic foundation surprisingly reduced the degrees of freedom compared with the existing method.

A bidomain model based BEM-FEM coupling formulation for anisotropic cardiac tissue.

A hybrid boundary element method (BEM)/finite element method (FEM) approach is proposed in order to properly consider the anisotropic properties of the cardiac muscle in the magneto- and electrocardiographic forward problem. Within the anisotropic myocardium a bidomain model based FEM formulation is applied. In the surrounding isotropic volume conductor the BEM is adopted. Coupling is enabled by requesting continuity of the electric potential and the normal of the current density across the boundary of the heart. Here, the BEM part is coupled as an equivalent finite element to the finite element stiffness matrix, thus preserving in part its sparse property. First, continuous convergence of the coupling scheme is shown for a spherical model comparing the computed results to an analytic reference solution. Then, the method is extended to the depolarization phase in a fibrous model of a dog ventricle. A precomputed activation sequence obtained using a fine mesh of the heart was downsampled and used to calculate body surface potentials and extracorporal magnetic fields considering the anisotropic bidomain conductivities. Results are compared to those obtained by neglecting in part or totally (oblique or uniform dipole layer model) anisotropic properties. The relatively large errors computed indicate that the cardiac muscle is one of the major torso inhomogeneities.

A finite element multigrid preconditioner for Chebyshev–collocation methods

This paper concerns the iterative solution of the linear system arising from the Chebyshev–collocation approximation of second-order elliptic equations and presents an optimal multigrid preconditioner based on alternating line Gauss–Seidel smoothers for the corresponding stiffness matrix of bilinear finite elements on the Chebyshev–Gauss–Lobatto grid.

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto-viscoplastic plane-strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto-viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto-plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non-hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.

A monotone finite element scheme for convection-diffusion equations

A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an M M -matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.

FINITE ELEMENTS FOR MODELLING BEAMS AFFECTED BY A DISTRIBUTED LOAD

Usually a finite element with cubic deflection approximation function is applied when evaluating the stress and strain field of bar structures. But such an element only approximately evaluates the actual strain field of the bar affected by a distributed load. The improved finite elements (Fig 1, 2) with fourth and fifth-order deflection approximation functions (1), (6) and (13) are presented in the actual manuscript. The fifth-order deflection approximation function is used for modelling the beams affected by a linearly distributed load (11). The plain bending of the finite element is modelled by 5 and 6 freedom degrees. The additional 5th and 6th freedom degrees are the deflection and deviation of the middle node of element (Fig 2). The element stiffness matrices (Table 1, 2) and node force vectors are presented. The created finite elements exactly modells the stress and strain field of bars, which are affected by distributed load, and also allow to compute directly the middle section displacements of bars. It creates conditions for diminishing the volume of problems and obtaining information, which is necessary to be analysed later. The reduced finite elements (Fig 4) are created by the elimination of the internal freedom degrees. Their number of freedom degrees is decreased up to the number of freedom degrees of a usually applied finite element. But the reduced finite elements have all afore-mentioned qualities. Formulas (20) and (21) are derived expressing the middle node displacements by the final node displacements. These formulas allow to compute the middle section displacements of the bar already after the solution of equation system. The proposed reduced elements can be introduced and applied in engineering practice very easily, because their stiffness matrix coincide with the stiffness matrix of a usual bar finite element. The created elements with internal freedom degrees are very important for the problems of structures optimization with displacement constraints, because the constraint of bar middle section displacement can form just in case, when this displacement is one of the problem's unknown. Also it is very important to decrease the number of unknowns of optimization problem.

Plate theory and complementary displacement method

With the complementary displacement method introduced in this article regarding plates, it is possible to deduce a constitutive law of a mechanical model from that of the usual three-dimensional (3-D) model. The model studied here is that of a plate undergoing infinitesimal transformations. The method is based on an appropriate kinematic formulation: it decomposes the field of displacements into the sum of a principal displacement, which allows the usual plate theory concepts to be introduced, and a complementary displacement which has been greatly neglected in the classical approach. The classical hypotheses, whether kinematic like those of Kirchhoff or Reissner, or static like that of plane constraints, are replaced here by the sole hypothesis that the laws of external forces are plate laws: the forces are independent of the complementary displacement and develop no virtual work in a virtual complementary displacement. This being assumed, we show the existence of a global constitutive law linking the fields of generalised deformations and generalised constraints, and also the identical nature of plate equilibrium and three-dimensional solid problems. A local constitutive law for plates can only be approached provided the plate is thin enough for its vast majority to be far enough from the edge. The most natural way is to assume that the field of generalised deformations is constant; hence the complementary displacement is the solution of an ordinary differential equation which, when solved, gives the constitutive law sought. The cases to which the equation is applied are an elastic isotropic material, producing classical results, and an elastic sandwich plate, leading to new results. Finally, when the field of generalised deformations is polynomial, the fields of complementary displacements and constraints can be found by polynomial identification, thus providing Saint-Venant polynomial solutions, as well as the stiffness matrix of a plate finite element.

Eigen-analysis using optimization with Rayleigh's quotient

An eigen-analysis is usually required if it is desired to obtain the natural frequencies and mode shapes (i.e. eigen-parameters) from a finite element mass and stiffness matrix. In certain instances, however, only the lowest and/or largest natural frequencies are of interest. It is shown that determining the lowest and largest eigen-parameters can be formulated as an unconstrained optimization with Rayleigh's quotient as the objective function. The Powell method and a heuristic search technique, called a genetic algorithm, are used in the optimization. In addition, four initial starting point estimators are discussed for the Powell method. The effectiveness of the proposed optimization is shown using a 6 and 8 degree-of-freedom system, and a 42 degree-of-freedom finite element model of a cantilevered Euler-Bernoulli beam. The lowest and largest eigen-parameters were shown to be extremely well identified. Furthermore, Powell's Method produced better results than the genetic algorithm in terms of efficiency and accuracy.

Micromechanically based stochastic finite elements: length scales and anisotropy

The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the finite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elements' stiffness matrices from the microstructures over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.

Exact calculation of natural frequencies of repetitive structures

Finite element stiffness matrix methods are presented for finding natural frequencies (or buckling loads) and modes of repetitive structures. The usual approximate finite element formulations are included, but more relevantly they also permit the use of 'exact finite elements', which account for distributed mass exactly by solving appropriate differential equations. A transcendental eigenvalue problem results, for which all the natural frequencies are found with certainty. The calculations are performed for a single repeating portion of a rotationally or linearly (in one, two or three directions) repetitive structure. The emphasis is on rotational periodicity, for which principal advantages include: any repeating portions can be connected together, not just adjacent ones; nodes can lie on, and members along, the axis of rotational periodicity; complex arithmetic is used for brevity of presentation and speed of computation; two types of rotationally periodic substructures can be used in a multi-level manner; multi-level non-periodic substructuring is permitted within the repeating portions of parent rotationally periodic structures or substructures and; all the substructuring is exact, i.e., the same answers are obtained whether or not substructuring is used. Numerical results are given for a rotationally periodic structure by using exact finite elements and two levels of rotationally periodic substructures. The solution time is about 500 times faster than if none of the rotational periodicity had been used. The solution time would have been about ten times faster still if the software used had included all the substructuring features presented.