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Tangent Stiffness Matrix (tangent + stiffness_matrix)
Selected AbstractsAnalytical stiffness matrices with Green,Lagrange strain measureINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005Pauli Pedersen Abstract Separating the dependence on material and stress/strain state from the dependence on initial geometry, we obtain analytical secant and tangent stiffness matrices. For the case of a linear displacement triangle with uniform thickness and uniform constitutive behaviour closed-form results are listed, directly suited for coding in a finite element program. The nodal positions of an element and the displacement assumption give three basic matrices of order three. These matrices do not depend on material and stress/strain state, and thus are unchanged during the necessary iterations for obtaining a solution based on Green,Lagrange strain measure. The approach is especially useful in design optimization, because analytical sensitivity analysis then can be performed. The case of a three node triangular ring element for axisymmetric analysis involves small modifications and extension to four node tetrahedron elements should be straight forward. Copyright © 2004 John Wiley & Sons, Ltd. [source] Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analysesINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2009F. Prunier Abstract The present paper investigates bifurcation analysis based on the second-order work criterion, in the framework of rate-independent constitutive models and rate-independent boundary-value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi-static boundary-value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9:203,274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207,220) and the plasticity limit criterion are revived in order to clarify the application field of the second-order work criterion and to contrast these criteria. The first part of this paper analyses the second-order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (d, = Md,) or in the dual form (d, = Nd,), is discussed. In the second part, the analysis is extended to the boundary-value problems in quasi-static conditions. Nonlinear 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 in the homogeneous and nonhomogeneous boundary-value problem. Copyright © 2008 John Wiley & Sons, Ltd. [source] An efficient co-rotational formulation for curved triangular shell elementINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2007Zhongxue Li Abstract A 6-node curved triangular shell element formulation based on a co-rotational framework is proposed to solve large-displacement and large-rotation problems, in which part of the rigid-body translations and all rigid-body rotations in the global co-ordinate system are excluded in calculating the element strain energy. Thus, an element-independent formulation is achieved. Besides three translational displacement variables, two components of the mid-surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out-of-plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co-rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd. [source] Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch testINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2004J. E. Dolbow Abstract We present a geometrically non-linear assumed strain method that allows for the presence of arbitrary, intra-finite element discontinuities in the deformation map. Special attention is placed on the coarse-mesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit. Given an underlying mesh and an arbitrary failure surface, we first construct an enriched approximation for the deformation map with the non-linear analogue of the extended finite element method (X-FEM). With regard to the richer space of functions spanned by the gradient of the enriched approximation, we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain. In particular, in those elements intersected by the failure surface, we construct enhanced strain approximations which are orthogonal to piecewise-constant stress fields. Contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields. Importantly, the present formulation gives rise to a symmetric tangent stiffness matrix, even in localized elements. The present modification also allows for the satisfaction of a discontinuous patch test, wherein two different constant stress fields (on each side of the failure surface) lie in the solution space. We demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface, particularly for nearly incompressible material response. Additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics. Copyright © 2003 John Wiley & Sons, Ltd. [source] Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2001Javier Bonet Abstract The paper discusses the problem of tension instability of particle-based methods such as smooth particle hydrodynamics (SPH) or corrected SPH (CSPH). It is shown that tension instability is a property of a continuum where the stress tensor is isotropic and the value of the pressure is a function of the density or volume ratio. The paper will show that, for this material model, the non-linear continuum equations fail to satisfy the stability condition in the presence of tension. Consequently, any discretization of this continuum will result in negative eigenvalues in the tangent stiffness matrix that will lead to instabilities in the time integration process. An important exception is the 1-D case where the continuum becomes stable but SPH or CSPH can still exhibit negative eigenvalues. The paper will show that these negative eigenvalues can be eliminated if a Lagrangian formulation is used whereby all derivatives are referred to a fixed reference configuration. The resulting formulation maintains the momentum preservation properties of its Eulerian equivalent. Finally a simple 1-D wave propagation example will be used to demonstrate that a stable solution can be obtained using Lagrangian CSPH without the need for any artificial viscosity. Copyright © 2001 John Wiley & Sons, Ltd. [source] Die shape design optimization of sheet metal stamping process using meshfree methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2001Nam Ho Kim Abstract A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity-based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum-based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation-dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring-back effect and to show the feasibility of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd. [source] A variational multiscale Newton,Schur approach for the incompressible Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2010D. Z. Turner Abstract In the following paper, we present a consistent Newton,Schur (NS) 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 NS 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. Copyright © 2009 John Wiley & Sons, Ltd. [source] Energy driven crack propagation at finite strains based on the embedded strong discontinuity approachPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009Radan Radulovic New advances in three-dimensional finite element modeling of crack propagation at finite strains are presented. The proposed numerical model is based on the Enhanced Assumed Strain concept. The enhanced part of the deformation gradient is associated with a displacement discontinuity. In contrast to previous works, a new, energy based criterion for crack propagation is presented. The necessity for a tracking algorithm for the crack path is avoided by using more than one discontinuity within each finite element. This leads to a strictly local formulation, i.e., no information about the neighboring elements are required. Further advantages of such a formulation are a symmetric tangent stiffness matrix and the reduction of locking effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | ||||||||||