Topic

# Tangent stiffness matrix

About: Tangent stiffness matrix is a(n) research topic. Over the lifetime, 1031 publication(s) have been published within this topic receiving 21140 citation(s).

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Abstract: It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative J 2 flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a class of non-associative flow rules are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton's method, whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.

1,626 citations

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Abstract: This paper describes a co-rotational formulation for three-dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted ‘strain measures’. The latter relate to standard beam theory but are embedded in a continuously rotating frame. A set of numerical examples show that the element provides an excellent numerical performance.

460 citations

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Abstract: This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the element independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.

351 citations

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TL;DR: It has been found that although the component of the stiffness matrix differentiating the enhanced stiffness model from the conventional one is not always positive definite, the resulting stiffness matrix can still be positive definite.

Abstract: This paper presents the enhanced stiffness modeling and analysis of robot manipulators, and a methodology for their stiffness identification and characterization. Assuming that the manipulator links are infinitely stiff, the enhanced stiffness model contains: 1) the passive and active stiffness of the joints and 2) the active stiffness created by the change in the manipulator configuration, and by external force vector acting upon the manipulator end point. The stiffness formulation not accounting for the latter is known as conventional stiffness formulation, which is obviously not complete and is valid only when: 1) the manipulator is in an unloaded quasistatic configuration and 2) the manipulator Jacobian matrix is constant throughout the workspace. The experimental system considered in this study is a Motoman SK 120 robot manipulator with a closed-chain mechanism. While the deflection of the manipulator end point under a range of external forces is provided by a high precision laser measurement system, a wrist force/torque sensor measures the external forces. Based on the experimental data and the enhanced stiffness model, the joint stiffness values are first identified. These stiffness values are then used to prove that conventional stiffness modeling is incomplete. Finally, they are employed to characterize stiffness properties of the robot manipulator. It has been found that although the component of the stiffness matrix differentiating the enhanced stiffness model from the conventional one is not always positive definite, the resulting stiffness matrix can still be positive definite. This follows that stability of the stiffness matrix is not influenced by this stiffness component. This study contributes to the previously reported work from the point of view of using the enhanced stiffness model for stiffness identification, verification and characterization, and of new experimental results proving that the conventional stiffness matrix is not complete and is valid under certain assumptions.

277 citations

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TL;DR: This paper proves that using the symmetric part of the tangent matrix, the Newton iteration retains its quadratic rate of convergence, and demonstrates that it is possible to analyze structures undergoing large rotations within a general co-rotational framework, using simple and economical finite elements.

Abstract: A systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis. This procedure is a generalization of the work presented in Computers and Structures, Volume 30, pp. 257–267. The basis of our approach is the element-independent co-rotational algorithm, where the element rigid body motion (translations and rotations) is separated from the deformational part of its total motion. The variation of this co-rotational relation results in a projector matrix, with the property that a consistent internal force vector is invariant under its action. The consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector. This results in an unsymmetric tangent matrix whose anti-symmetric part depends on the out-of-balance force vector. In this paper we prove that using the symmetric part of the tangent matrix, the Newton iteration retains its quadratic rate of convergence. This approach has been used to solve a number of large rotation test example problems. The results demonstrate that it is possible to analyze structures undergoing large rotations within a general co-rotational framework, using simple and economical finite elements. The resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre- and post-processors to the element routines.

260 citations