Tangent stiffness matrix

About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.

Material stability analysis based on the local and global elasto-plastic tangent operators

Abstract: The present paper investigates bifurcation in geomaterials with the help of the second-order work criterion. The approach applies mainly to non associated materials such as soils. The analysis usually performed at the material point level is extended to quasi-static boundary value problems, by considering the finite element stiffness matrix. The first part of the paper reminds some results obtained at the material point level. The bifurcation domain is presented in the 3D principal stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. In the second part, the analysis is extended to boundary value problems in quasi-static conditions. Non-linear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode even for non homogeneous boundary value problems.

Stiff arc length constraint in nonlinear FEA

Abstract: Arc length constraints enable iterative solution procedures in nonlinear finite element analysis to converge even at critical points. The arc length constraint replaces the conventional m×m stiffness matrix with an augmented (m+1)×(m+1) stiffness matrix. Its use is referred to as arc length control, in contrast to load control which furnishes the conventional stiffness matrix. In the current article, an apparently new arc length constraint is introduced. It identifies arc length parameters maximizing the stiffness (absolute value of the determinant) of the augmented matrix. The parameters, viewed as a vector, must be perpendicular to the rows of the stiffness matrix, likewise considered vectors. The augmented stiffness matrix is nonsymmetric and lacks the small bandwidth of the conventional stiffness matrix. However, using a block triangularization, it is demonstrated that a solution may be attained by standard finite element operations, namely triangularization of a banded nonsingular portion of the stiffness matrix followed by forward and backward substitutions involving banded lower and upper triangular matrices. The proposed constraint is expected to permit convergence under longer arc lengths than currently implemented methods. A simple example is given illustrating the application of the constraint.

A Tetrahedral Decomposition Method for Computing Tangent Curves of 3D Vector Fields

Abstract: This paper presents the development of certain highly efficient and accurate method for computing tangent curves for three-dimensional vector fields. Unlike conventional methods, such as Runge-Kutta method, for computing tangent curves which produce only approximations, the method developed herein produces exact values on the tangent curves based upon piecewise linear variation over a tetrahedral domain in 3D. This new method assumes that the vector field is piecewise linearly defined over a tetrahedron in 3D domain. It is also required to decompose the hexahedral cell into five or six tetrahedral cells for three-dimensional vector fields. The critical points can be easily found by solving a simple linear system for each tetrahedron. This method is to find exit points by producing a sequence of points on the curve with the computation of each subsequent point based on the previous. Because points on the tangent curves are calculated by the explicit solution for each tetrahedron, this new method provides correct topology in visualizing 3D vector fields.

Finite element formulation of Timoshenko tapered beam-column element for large displacement analysis based on the exact shape functions

Abstract: ABSTRACT A numerical formulation was carried out in this paper to produce the tangent stiffness matrix for two-nodal tapered Timoshenko beam-column elements for geometrically nonlinear analysis. The proposed solution is based on the exact shape functions and their derivatives describing the non-uniformity of the element properties. The section properties were presented as exponential functions with tapering indices to illustrate the variations in section properties along the tapered element length. The model is applicable for elements with different solid and hollow cross-sections. The proposed formulation is embedded into a Visual Basic code to carry out the analysis accompanied by many examples for validating its accuracy and efficiency. The model results are compared with those of commercial software and cited references that showed high accurate results with a small number of elements.