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.

A variational multiscale Newton-Schur approach for the incompressible Navier-Stokes equations

Abstract: In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices. In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and fine-scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows.

The lateral buckling of timber arches

Abstract: This paper presents the stability analyses of glulam arches subjected to distributed vertical loading. The present analysis employs a strain-based formulation of a nonlinear geometrically exact three-dimensional beam theory. The influence of the relative height of the arch on the lateral buckling load is studied. The buckling load is determined by bisection method with observing the sign of the determinant of the tangent stiffness matrix. The post-critical load deflection path is traced by a modified arc–length method. Such influences are shown for arches with a constant cross-section or constant volume. After determining the most favorable height of the arch, the influence of the number and position of lateral supports is shown. We also compare the deflections, bending, and radial stresses at the lateral buckling states to the limit values which are recommended by European standards.

A General Formulation for the Stiffness Matrix of Parallel Mechanisms

Abstract: Starting from the definition of a stiffness matrix, the authors present a new formulation of the Cartesian stiffness matrix of parallel mechanisms. The proposed formulation is more general than any other stiffness matrix found in the literature since it can take into account the stiffness of the passive joints, it can consider additional compliances in the joints or in the links and it remains valid for large displacements. Then, the validity, the conservative property, the positive definiteness and the relation with other formulations of stiffness matrices are discussed theoretically. Finally, a numerical example is given in order to illustrate the correctness of this matrix.

Geometrically nonlinear formulation of Generalized Beam Theory for the analysis of thin-walled circular pipes

Abstract: In this paper, the formulation of Generalized Beam Theory (GBT) for the geometrically nonlinear analysis of thin-walled circular pipes is presented. GBT is a computationally efficient numerical method which is especially formulated for thin-walled members with a capacity of determining the cross-sectional deformation through a combination of a set of pre-determined cross-sectional deformation modes. In this study, the current GBT analysis of circular pipes which is limited to buckling analysis is enhanced to a full geometrically nonlinear analysis. This new formulation considers the nonlinear membrane kinematic description based on the Green–Lagrange strain definition. The nonlinear tangent stiffness matrices and the internal force vectors are derived from the variation of the internal energy which results in third and fourth-order GBT deformation mode coupling tensors. These tensors can predetermine the type of GBT deformation modes needed for the nonlinear analysis based on the applied loading conditions. In addition to the classical GBT deformation modes, the non-conventional GBT deformations modes have a vital role since without these modes the coupling tensors and the nonlinear stiffness matrix related to the transverse and the shear membrane energy will be lost. Here, to illustrate the application and capabilities of the developed GBT formulation, two numerical examples involving transverse and longitudinal bending are presented to show the nonlinear relationship between bending, cross-sectional ovalization, and higher local deformation modes. For the purpose of validation, these examples are compared with refined shell finite element models in both displacement and stress fields.