scholarly journals Recursive calculation of curved finite elements stiffness matrices

1984 ◽  
Vol 10 (4-5) ◽  
pp. 387-397 ◽  
Author(s):  
M.L. Baart ◽  
R.J.Y. McLeod
Keyword(s):  
Finite Elements ◽  
Stiffness Matrices ◽  
Recursive Calculation

scholarly journals Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping

2007 ◽  
Vol 17 (3) ◽  
pp. 375-393 ◽  
Author(s):  
Sébastien Jund ◽  
Stéphanie Salmon
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Finite Elements ◽  
Taylor Expansion ◽  
High Order ◽  
Computational Time ◽  
Mass Lumping ◽  
Stiffness Matrices ◽  
High Order Time ◽  
Finite Element Schemes

Arbitrary High-Order Finite Element Schemes and High-Order Mass LumpingComputers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elements of orderkand show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up toP5elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.

Finite Element Modeling of Dynamic Behavior of Some Basic Structural Members

1992 ◽  
Vol 114 (1) ◽  
pp. 3-9 ◽  
Author(s):  
R. C. Engels
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Modeling ◽  
Mass Matrix ◽  
Matrix Approach ◽  
Structural Members ◽  
Shape Functions ◽  
Generalized Coordinates ◽  
Stiffness Matrices ◽  
Consistent Mass Matrix

A method is described to model the dynamics of finite elements. The assumed modes method is used to show how static shape functions approximate the element mass distribution. The deterioration of the modal content of a model can be linked to the neglect of interface restrained assumed modes. Restoration of a few of these modes leads to higher accuracy with fewer generalized coordinates compared to the standard consistent mass matrix approach. Also, no need exists for subdivision of basic elements such as rods and beams. The mass and stiffness matrices for several basic elements are derived and used in demonstration problems.

scholarly journals The Inf-Sup Condition Tests for Shell/Plate Finite Elements / Testy Warunku Inf-Sup Do Oceny Płytowych I Powłokowych Elementów Skonczonych

2011 ◽  
Vol 57 (4) ◽  
pp. 425-447 ◽  
Author(s):  
W. Gilewski ◽  
M. Sitek
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
High Performance ◽  
Point Of View ◽  
Numerical Verification ◽  
Shell Elements ◽  
Numerical Tests ◽  
Stiffness Matrices ◽  
Finite Element Technology ◽  
Selection Of

Abstract Development of high-performance finite elements for thick, moderately thick, as well as thin shells and plates, was one of the active areas of the finite element technology for 40 years, followed by hundreds of publications. A variety of shell elements exist in the FE codes, but “the best” finite element is still to be discovered. The paper deals with an evaluation of some existing shell finite elements, from the point of view of the third of three requirements to be satisfied by the element: ellipticity, consistency and inf-sup condition. It is difficult to prove the inf-sup condition analytically, so, a numerical verification is proposed. A set of numerical tests is considered for shell and plate problems. Two norm matrices and a selection of the stiffness matrices (bending, shear and membrane dominated) are analysed. Finite elements from various computer systems can be evaluated and compared with the use of the proposed tests.

Closed-form stiffness matrices for the linear strain and quadratic strain tetrahedron finite elements

1992 ◽  
Vol 45 (2) ◽  
pp. 237-242 ◽  
Author(s):  
P.S. Shiakolas ◽  
R.V. Nambiar ◽  
K.L. Lawrence ◽  
W.A. Rogers
Keyword(s):  
Finite Elements ◽  
Closed Form ◽  
Linear Strain ◽  
Stiffness Matrices

Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Brian Tinsley ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Elements ◽  
Computer Aided Design ◽  
Eigenvalue Analysis ◽  
Reference Configuration ◽  
Position Vector ◽  
B Spline ◽  
Split Method ◽  
Stiffness Matrices ◽  
General Continuum ◽  
Aided Design

Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.

Simple and Efficient Tetrahedral Finite Elements With Rotational Degrees of Freedom for Solid Modeling

2007 ◽  
Vol 7 (4) ◽  
pp. 382-393 ◽  
Author(s):  
X. Hua ◽  
C. W. S. To
Keyword(s):  
Finite Elements ◽  
Degrees Of Freedom ◽  
Benchmark Problems ◽  
Hybrid Strain ◽  
Tetrahedral Elements ◽  
Mixed Variational Principle ◽  
Stiffness Matrices ◽  
Rigid Body Modes ◽  
Rotational Degrees Of Freedom ◽  
Six Dof

A mixed variational principle and derivation of two simple and efficient tetrahedral finite elements with rotational degrees of freedom (DOF) are presented. Each element has four nodes. Every node has six DOF, which include three translational and three rotational DOF. Each element is capable of providing six rigid-body modes. The rotational DOF are based on the displacement formulation, while the translational DOF are hinged on the hybrid strain Hellinger–Reissner functional. Explicit expressions for stiffness matrices are obtained. Element performance has been evaluated with benchmark problems, indicating that they have superior accuracy compared with other lower-order tetrahedral elements.

scholarly journals FINITE ELEMENT SOLUTION OF MULTI-SCALE TRANSPORT PROBLEMS USING THE LEAST SQUARES-BASED BUBBLE FUNCTION ENRICHMENT

2012 ◽  
Vol 03 (04) ◽  
pp. 1250019
Author(s):  
A. YAZDANI ◽  
V. NASSEHI
Keyword(s):  
Finite Elements ◽  
Least Squares ◽  
Computational Cost ◽  
Shape Functions ◽  
Element Level ◽  
Polynomial Coefficients ◽  
Stiffness Matrices ◽  
Standard Linear ◽  
Bubble Functions ◽  
Transport Problems

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements, employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients. The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares, resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values. The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred. Furthermore, the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions. It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes. On the other hand, it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

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