Analytic formulation of stiffness matrix for axisymmetric solids

Stiffness matrices for axisymmetric solids with arbitrary loading were derived through the application of variational principles in analytic form. The singularities in the stiffness matrices were removed through displacement constraints along the axis of symmetry for each circumferential mode. The shear stresses and maximum deflections of a set of Saint-Venant flexural problems were obtained both analytically and numerically. The results indicate that the finite element analysis with the analytic stiffness matrix provides a very good solution. The same problems were solved with a commercial code, ANSYS. and showed that the analytic stiffness matrix contributed to a faster convergence rate as the number of elements increases in an analysis.     

Curved beam element stiffness matrix formulation

Stiffness matrix of order 12 × 12, for a curved beam element has been formulated involving all the forces together, using Castigliano's theorem. Effects of transverse shear forces and tangential thrust are also taken into account. In earlier works, stiffness submatrices for the two uncoupled systems of forces are formulated independently and then they are combined to give the overall 12 × 12 matrix. A program subroutine, NEWCBM for the stiffness matrix formulation of curved beams has been written in FORTRAN which can be added to the element library of general purpose computer programs like SAP-IV and its improved versions. Example problems have been worked to check the accuracy of this formulation.     

The natural factor formulation of the stiffness for the matrix displacement method

The natural formulation of the elemental stiffness has proved a most significant interpretation of the elastic behaviour of an element, both from the physical and mathematical point of view. It yields not only concise and elegant matrix expressions for the elements — especially for simplex elements — but leads effortlessly to the non-linear effects associated with large displacements. In the present paper the basic idea is brought to a logical conclusion by the direct construction of the factorised form of the elemental and overall stiffness matrices from their natural stiffness roots. This technique is denoted as the natural factor formulation. It is believed that the new approach leads to a better understanding of the relevant error analysis so powerfully initiated by Rosanoff et al.; see e.g. [20]. The natural factor formulation may also be shown to retain more numerical information of the idealised physical system [21, 28]. Both static and dynamic problems are considered here. For additional information the reader is referred to [16].     

The tangent stiffness matrix for an absolute interface coordinates floating frame of reference formulation

Multibody System Dynamics - In this work, a full and complete development of the tangent stiffness matrix is presented, suitable for the use in an absolute interface coordinates floating frame of...     

Closed form stiffness matrix solutions for some commonly used hybrid finite elements

Closed form solutions for the element stiffness matrices for four commonly used hybrid finite elements, namely, the 5-β-I, 5-β-II, RGH4 and RGH8 elements are given in detail. By using the closed forms, the computational effort needed for the formation of the element stiffness matrices is greatly reduced and the danger of matrix rank deficiency due to the use of insufficient integration points is eliminated. Numerical experiment shows that the CPU time needed for the element stiffness matrix formation procedure for hybrid elements is much less than that for conventional high order displacement finite elements while the quality of the solutions obtained is of similar, and in many cases of better, accuracy.     

A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements

This study presents a new algorithm developed in order to remove instabilities observed in the simulation of unsteady viscoelastic fluid flows in the framework of the spectral element method. In this study, we consider a particular model of the finite extensible nonlinear elastic family, FENE-P, but the method could be applied to other differential constitutive equations. Two distinct constraints for the FENE-P equation are imposed: (i) the square of the corresponding finite extensibility parameter of the polymer must be an upper limit for the trace of the conformation tensor and (ii) the eigenvalues of the conformation tensor should remain positive at all steps of the simulation. Negative eigenvalues cause the unbounded growth of instabilities in the flow. The proposed transformation is an extension of the matrix logarithm formulation originally presented by Fattal and Kupferman [1] and [2]. To evaluate the capability of this new algorithm with the classical conformation tensor, comprehensive studies have been done based on the linear stability analysis to show the influence of this method on the resulting eigenvalue spectra and explain its success to tackle high Weissenberg numbers. With this new method one can tackle high Weissenberg number flow at values of practical interest. A neat improvement of the computational algorithm with stable convergence has been demonstrated in this study.     

A FORTRAN routine for computation of coupled bending-torsional dynamic stiffness matrix of beam elements

A FORTRAN subroutine for computing coupled bending-torsional dynamic stiffness matrix of a uniform beam element is developed. An annotated listing of the program is presented. The application of the subroutine is discussed with particular reference to an established algorithm. An illustrative example on the use of the subroutine is given with representative results.     

A matrix formulation for the estimation of weighting sequence matrix

The use of psoudo-random binary sequence for the estimation of weighting sequence matrices of multivariable discrete-time systems is studied using the matrix approach and the least squares method. The input phase separation scheme is used for test signal. Another approach, which is a little less optimal but yet computationally much simpler, is also presented.     

Geometrically non-linear formulation of flexible multibody systems in terms of beam elements: Geometric stiffness

The occurrence of strong deflections and major axial forces in many applications involving flexible multibodies entails including non-linear terms coupling deformation-induced axial and transverse displacements in the motion equation. The formulations, including such terms, are known as geometrically non-linear formulations. The authors have developed one such formulation that preserves higher-order terms in the strain energy function. By expressing such terms as a function of selected elastic coordinates, three stiffness matrices and two non-linear vectors of elastic forces are defined. The first matrix is the conventional constant-stiffness matrix, the second is the classical geometric stiffness matrix and the third is a second-order geometric stiffness matrix. The aim of this work is to define the third matrix and the two non-linear vectors of elastic forces by using the finite-element method.     

A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation

Finite element analysis using plate elements based on the absolute nodal coordinate formulation (ANCF) can predict the behaviors of moderately thick plates subject to large deformation. However, the formulation is subject to numerical locking, which compromises results. This study was designed to investigate and develop techniques to prevent or mitigate numerical locking phenomena. Three different ANCF plate element types were examined. The first is the original fully parameterized quadrilateral ANCF plate element. The second is an update to this element that linearly interpolates transverse shear strains to overcome slow convergence due to transverse shear locking. Finally, the third is based on a new higher order ANCF plate element that is being introduced here. The higher order plate element makes it possible to describe a higher than first-order transverse displacement field to prevent Poisson thickness locking. The term “higher order” is used, because some nodal coordinates of the new plate element are defined by higher order derivatives. The performance of each plate element type was tested by (1) solving a comprehensive set of small deformation static problems, (2) carrying out eigenfrequency analyses, and (3) analyzing a typical dynamic scenario. The numerical calculations were made using MATLAB. The results of the static and eigenfrequency analyses were benchmarked to reference solutions provided by the commercially available finite element software ANSYS. The results show that shear locking is strongly dependent on material thickness. Poisson thickness locking is independent of thickness, but strongly depends on the Poisson effect. Poisson thickness locking becomes a problem for both of the fully parameterized element types implemented with full 3-D elasticity. Their converged results differ by about 18 % from the ANSYS results. Corresponding results for the new higher order ANCF plate element agree with the benchmark. ANCF plate elements can describe the trapezoidal mode; therefore, they do not suffer from Poisson locking, a reported problem for fully parameterized ANCF beam elements. For cases with shear deformation loading, shear locking slows solution convergence for models based on either the original fully parameterized plate element or the newly introduced higher order plate element.     

An optimized block matrix manipulation for boundary elements with subregions

This work discusses the subregions technique and presents a boundary element formulation with an optimized vector generation for the system of equations.The procedure only requires storage for a minimum number of non-zero matrix coefficients plus the possibly non-zero ones which appear when a maximum pivot selection is activated during a Gauss elimination routine. The technique produces great savings in computer run time and is implemented in a 3-D modular elasticity code with quadrilateral and triangular elements, continuous and discontinuous, constant, linear and quadratic. The FORTRAN computer listing of the main routines developed is included.     

An efficient formulation of hexahedral elements with high accuracy for bending and incompressibility

The Hu-Washizu variational principle is used to formulate hexahedral elements so that these elements exhibit high coarse-mesh accuracy for problems involving bending of compressible or incompressible materials. An orthogonalization procedure is used on the displacement-gradient, strain, and stress fields so that the resulting element stiffness does not require any matrix inversions. Results are given for several problems which demonstrate excellent performance for coarse and refined meshes.     

Economical stiffness formulations for nonlinear finite elements

Element-level calculations often represent a significant part of the computing effort in a nonlinear finite element solution, especially when three-dimensional, higher-order elements are used. This paper explores some possibilities for increasing the efficiency of element computations within the general framework of a Newton-Raphson solution technique. A modified tangent stiffness formulation is introduced which provides relatively fast convergence without the extreme computational effort sometimes associated with the usual Newton-Raphson interaction. Numerical examples are used to illustrate the behavior of the method. The use of different types of element formulations within a single finite element mesh, according to the expected or observed degree of nonlinearity, is also identified as a means of reducing solution cost.     

Geometrically nonlinear formulation for the axi-symmetric transition finite elements

This paper presents a geometrically nonlinear formulation for the axi-symmetric transition finite elements using total lagrangian approach. The basic element is formulated using properties of the axi-symmetric solids and the axi-symmetric shells. A novel feature of the formulation presented here is that the restriction on the magnitude of the rotations for the shell nodes of the transition element is eliminated. This is accomplished by retaining true nonlinear functions of nodal rotations in the definition of the element displacement field. Such transition elements are essential for geometrically nonlinear applications requiring both axi-symmetric solids and the axi-symmetric shells. They ensure proper connection of the axi-symmetric solid portion of the structure to the shell like portion of the structure. It is shown that the selection of different stress and strain components at the integration points does not effect the overall linear response of the element. However, in the geometrically nonlinear formulation, it is necessary to select appropriate components of the stresses and the strains at the integration point for accurate and converging element behavior. Numerical examples are presented to demonstrate such characteristics of the transition elements.     

Robust alternative minimization for matrix completion

Recently, much attention has been drawn to the problem of matrix completion, which arises in a number of fields, including computer vision, pattern recognition, sensor network, and recommendation systems. This paper proposes a novel algorithm, named robust alternative minimization (RAM), which is based on the constraint of low rank to complete an unknown matrix. The proposed RAM algorithm can effectively reduce the relative reconstruction error of the recovered matrix. It is numerically easier to minimize the objective function and more stable for large-scale matrix completion compared with other existing methods. It is robust and efficient for low-rank matrix completion, and the convergence of the RAM algorithm is also established. Numerical results showed that both the recovery accuracy and running time of the RAM algorithm are competitive with other reported methods. Moreover, the applications of the RAM algorithm to low-rank image recovery demonstrated that it achieves satisfactory performance.     

A general formulation of structural topology optimization for maximizing structural stiffness

This paper presents a general formulation of structural topology optimization for maximizing structure stiffness with mixed boundary conditions, i.e. with both external forces and prescribed non-zero displacement. In such formulation, the objective function is equal to work done by the given external forces minus work done by the reaction forces on prescribed non-zero displacement. When only one type of boundary condition is specified, it degenerates to the formulation of minimum structural compliance design (with external force) and maximum structural strain energy design (with prescribed non-zero displacement). However, regardless of boundary condition types, the sensitivity of such objective function with respect to artificial element density is always proportional to the negative of average strain energy density. We show that this formulation provides optimum design for both discrete and continuum structures.     

Exact stiffness matrix for beams on elastic foundation

An exact stiffness matrix of a beam element on elastic foundation is formulated. A single element is required to exactly represent a continuous part of a beam on a Winkler foundation. Thus only a few elements are sufficient for a typical problem solution. The stiffness matrix is assembled in a computer program and some numerical examples are presented.     

Kernel ICA: An alternative formulation and its application to face recognition

This paper formulates independent component analysis (ICA) in the kernel-inducing feature space and develops a two-phase kernel ICA algorithm: whitened kernel principal component analysis (KPCA) plus ICA. KPCA spheres data and makes the data structure become as linearly separable as possible by virtue of an implicit nonlinear mapping determined by kernel. ICA seeks the projection directions in the KPCA whitened space, making the distribution of the projected data as non-gaussian as possible. The experiment using a subset of FERET database indicates that the proposed kernel ICA method significantly outperform ICA, PCA and KPCA in terms of the total recognition rate.     

Composite finite elements for problems containing small geometric details

In this paper, we will present efficient strategies how composite finite elements can be realized for the discretization of PDEs on domains containing small geometric details. In contrast to standard finite elements, the minimal dimension of this new class of finite element spaces is completely independent of the number of geometric details of the physical domains. Hence, it allows coarse level discretization of PDEs which can be used, e.g., preferably for multi-grid methods and homogenization of PDEs in non-periodic situations. Received: 23 September 1996 / Accepted: 23 January 1997