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 Tangent Stiffness MLPG Method for Atom/Continuum Multiscale Simulation

Abstract: The main objective of this paper is to develop a multiscale method for the static analysis of a nano-system, based on a combination of molecular mechanics and MLPG methods. The tangent-stiffness formulations are given for this multiscale method, as well as a pure molecular mechanics method. This method is also shown to naturally link the continuum local balance equation with molecular mechanics, directly, based on the stress or force. Numerical results show that this multiscale method quite accurate. The tangent-stiffness MLPG method is very effective and stable in multiscale simulations. This multiscale method dramatically reduces the computational cost, but it still can provide reasonable accuracy in some regions of the model. keyword: Molecular mechanics, Multiscale method, continuum mechanics, MLPG.

Stability analysis of thin-walled beams with open section subject to arbitrary loads

Abstract: In this paper, we present an original work for the examination of the stability of thin-walled beams with open section subject to arbitrary loads. It is based on a 3D nonlinear model where the equilibrium and material constitutive equations are established without any assumption on the torsion angle amplitude which leads to strong nonlinearity. In a recently published article [1] , we compared this model without simplification to three others models with cubic, quadratic and linear simplifications. The efficiency of the model is confirmed from benchmark solutions. For this reason, we propose in this work to use this model without simplification in the presence of external forces. When these external forces are eccentric they make the nonlinear problem very difficult to solve because the tangent stiffness matrix depends on the load. In presence of arbitrary loads and large torsion context, the right hand side of the equilibrium equations is highly nonlinear and contributes to the tangent stiffness matrix. For this purpose, we use a continuation algorithm based on the Asymptotic Numerical Method ANM, recently published by the authors in [2] . The ANM is a computational tool for solving nonlinear equations numerically; this is achieved by associating the finite element method and a Taylor series expansions technique without any correction and iteration steps. By this way, we compute a large part of the branch by inverting only one stiffness matrix. The efficiency of the present extended model is tested on original applications of open sections thin-walled beams under arbitrary and eccentric loads. A comparison of the obtained results with those computed by Abaqus industrial code is presented.

Efficient numerical method in second-order inelastic analysis of space trusses

Abstract: In this paper, the Newton-Raphson method is combined with three different algorithms. These algorithms are the generalized minimum residual (GMRES), the least squares (LSQR), and the biconjugate gradient (BCG). Of these algorithms, the most effective at reducing the number of iterations and the time required is identified. A common characteristic of these algorithms is that they replace the inversion of the tangent stiffness matrix with an iterative procedure to solve the linearized system of equations. A computer program based on three algorithms is developed to numerically solve a system of nonlinear equations. The procedure can be applied to analysis of structures with complex behaviors, including unloading, snap-through buckling, and inelastic postbuckling analyses. To demonstrate the efficiency and accuracy of the method developed here, some well-known trusses are investigated and analyzed using the various aforementioned algorithms. Results show that the biconjugate gradient algorithm is a m...

Sensitivity analysis of bifurcation load of finite-dimensional symmetric systems

Abstract: Three methods are presented for sensitivity analysis of bifurcation load factor of finite-dimensional conservative symmetric systems subjected to a set of symmetric proportional loads. In the first method, a conventional method with diagonalization is utilized to derive an explicit formula of sensitivity coefficients corresponding to a minor imperfection. Next, a new concept is introduced to find the sensitivity coefficients of the load factor, displacements and the eigenmodes under fixed lowest eigenvalue of the tangent stiffness matrix. Based on this concept, a method is presented for finding approximate sensitivity coefficients of the buckling load factor. Finally, a direct method is presented to find the accurate sensitivity coefficients of the bifurcation load factor, displacements at buckling and the buckling mode of a symmetric system. Note that different formula should be used for sensitivity analysis of a limit point load factor. In the examples, the proposed three methods are compared in view of accuracy of the results and simplicity in coding.