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Tangent Matrix (tangent + matrix)

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Engineering100%

Selected Abstracts

Consistent tangent matrices for density-dependent finite plasticity models

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 11 2001
Agustí Pérez-Foguet
Abstract The consistent tangent matrix for density-dependent plastic models within the theory of isotropic multiplicative hyperelastoplasticity is presented here. Plastic equations expressed as general functions of the Kirchhoff stresses and density are considered. They include the Cauchy-based plastic models as a particular case. The standard exponential return-mapping algorithm is applied, with the density playing the role of a fixed parameter during the nonlinear plastic corrector problem. The consistent tangent matrix has the same structure as in the usual density-independent plastic models. A simple additional term takes into account the influence of the density on the plastic corrector problem. Quadratic convergence results are shown for several representative examples involving geomaterial and powder constitutive models. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
C. Miehe
Abstract The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that ,-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd. [source]

A new damage model based on non-local displacements

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 5 2005
Antonio Rodríguez-Ferran
Abstract A new non-local damage model is presented. Non-locality (of integral or gradient type) is incorporated into the model by means of non-local displacements. This contrasts with existing damage models, where a non-local strain or strain-related state variable is used. The new model is very attractive from a computational viewpoint, especially regarding the computation of the consistent tangent matrix needed to achieve quadratic convergence in Newton iterations. At the same time, its physical response is very similar to that of the standard models, including its regularization capabilities. All these aspects are discussed in detail and illustrated by means of numerical examples. Copyright © 2005 John Wiley & Sons, Ltd. [source]

An unconditionally convergent algorithm for the evaluation of the ultimate limit state of RC sections subject to axial force and biaxial bending

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2007
G. Alfano
Abstract We present a numerical procedure, based upon a tangent approach, for evaluating the ultimate limit state (ULS) of reinforced concrete (RC) sections subject to axial force and biaxial bending. The RC sections are assumed to be of arbitrary polygonal shape and degree of connection; furthermore, it is possible to keep fixed a given amount of the total load and to find the ULS associated only with the remaining part which can be increased by means of a load multiplier. The solution procedure adopts two nested iterative schemes which, in turn, update the current value of the tentative ultimate load and the associated strain parameters. In this second scheme an effective integration procedure is used for evaluating in closed form, as explicit functions of the position vectors of the vertices of the section, the domain integrals appearing in the definition of the tangent matrix and of the stress resultants. Under mild hypotheses, which are practically satisfied for all cases of engineering interest, the existence and uniqueness of the ULS load multiplier is ensured and the global convergence of the proposed solution algorithm to such value is proved. An extensive set of numerical tests, carried out for rectangular, L-shaped and multicell sections shows the effectiveness of the proposed solution procedure. Copyright © 2007 John Wiley & Sons, Ltd. [source]

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

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2010
D. 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]

Description for smooth contact conditions based on the internal geometry of contact surfaces

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Alexander Konyukhov Dr.
A kinematical approach, based on the consideration of the contact conditions in the local coordinate system, is proposed for the contact description and for consistent linearization. This leads to a simple structure of the tangent matrix, which is subdivided into main, rotational and curvature parts. Various alternatives neglecting parts of the contact tangent matrix are considered. Representative examples show the effectiveness of the proposed approach for contact problems with arbitrary large deformation. [source]