Numerical aspects of microplane constitutive models for concrete

This paper presents a comprehensive study on the numerical aspects of a class of microplane constitutive models for concrete, with emphasis on the most popular one, i.e., the model M4 developed by Bažant and co-workers. The effect of the computational procedures for the microplane shear stress components, the strain increment magnitude, the integration scheme and the loading direction on the model responses are investigated in detail. Several problems in the responses of the model, from the computational point of view, are detected and discussed. Some procedures to enhance the numerical performance of the model are then proposed. These include a removal of the tensile volumetric stress boundary in the original formulation, the introduction of a novel semi-explicit numerical algorithm for the microplane volumetric and deviatoric stresses, and a method to minimise the sensitivity of the model responses to the loading direction through a simple meshing-based integration scheme. Finally, some practical considerations for the choice of numerical integration scheme in the finite element calculations of large-scale concrete structures with the microplane model M4 are presented.     

A comparison of two microplane constitutive models for quasi-brittle materials

This article presents a comparison of two microplane constitutive models. The basis of the microplane constitutive models are described and the adopted assumptions for the conception of these models are discussed, with regard to: decomposition of the macroscopic strains into the microplanes, definition of the microplane material laws, including the choice of variables that control the material degradation, and homogenization process to obtain the macroscopic quantities. The differences between the two models, with respect to the employed assumptions, are emphasized and expressions to calculate the macroscopic stresses are presented. The models are then used to describe the behavior of quasi-brittle materials by finite element simulations of uniaxial tension and compression and pure share stress tests. The results of the simulations permit to compare the capability of the models in describing the post critical strain-softening behavior, without numerically induced strain localization.     

A regularization approach for damage models based on a displacement gradient

Common material models that take into account softening effects due to damage encounter the problem of ill-posed boundary value problems if no regularization is applied. This condition leads to a non-unique solution for the resulting algebraic system and a strong mesh dependence of the numerical results. A possible solution approach to prevent this problem is to apply regularization techniques that take into account the non-local behavior of the damage [1]. For this purpose a field function is used to couple the local damage parameter to a non-local level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized. In contrast, we present a novel approach to regularization in which no field function is needed [2]. Hereto, the regularization is carried out by means of the divergence of the displacements and no additional quantity is necessary since the displacements are already defined on a non-local level. The idea is that with an increasing value of the damage the element's volume will increase as well. This is a result of the softening due to the occurring damage. The increasing volume can be measured by the divergence of the displacements which can be penalized by an additional energy part. The lack of any field function and the regularization by the use of the divergence of the displacements entails several numerical advantages: the computational effort is considerably reduced and the convergence behavior is improved as well. Naturally, the numerical results are mesh independent due to the regularization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On quasistatic inelastic models of gradient type with convex composite constitutive equations

This article defines and presents the mathematical analysis of a new class of models from the theory of inelastic deformations of metals. This new class, containing so called convex composite models, enlarges the class containing monotone models of gradient type defined in [1]. This paper starts to establish the existence theory for models from this new class; we restrict our investigations to the coercive and linear self-controlling case.     

A variational coupled damage-plasticity model via gradient enhancement of the free energy function

We present a coupled damage-plasticity model, whose regularization is achieved through gradient enhancement of the free energy function by introducing new variables. They serve to transport the values of the inelastic variables across the element boundaries, and their gradients play regularisation role in the model. Variational formulation results in a pure minimisation problem, leading to model non-local in nature and preserving C⁰ interpolation order of the variables. Numerical examples illustrating the performance of the proposed model are presented. It is shown that the pathological mesh dependence is efficiently removed, together with the difficulties of numerical calculations in the softening range. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the truncated conjugate gradient method

In this paper, we consider the truncated conjugate gradient method for minimizing a convex quadratic function subject to a ball trust region constraint. It is shown that the reduction in the objective function by the solution obtained by the truncated CG method is at least half of the reduction by the global minimizer in the trust region. Received January 19, 1999 / Revised version received October 1, 1999?Published online November 30, 1999     

On the convergence of conjugate gradient algorithms

In this paper we present a new family of conjugate gradientalgorithms. This family originates in the algorithms providedby Wolfe and Lemaréchal for non-differentiable problems.It is shown that the Wolfe-Lemaréchal algorithm is identicalto the Fletcher-Reeves algorithm when the objective functionis smooth and when line searches are exact. The convergenceproperties of the new algorithms are investigated. One of themis globally convergent under minimum requirements on the directionalminimization.     

On the convergence of the projected gradient method

We study the projected gradient algorithm for linearly constrained optimization. Wolfe (Ref. 1) has produced a counterexample to show that this algorithm can jam. However, his counterexample is only ¹( ⁿ ), and it is conjectured that the algorithm is convergent for ²-functions. We show that this conjecture is partly right. We also show that one needs more assumptions to prove convergence, since we present a family of counterexamples. We finally give a demonstration that no jamming can occur for quadratic objective functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada     

On the inexact scaled gradient projection method

Computational Optimization and Applications - The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense....     

On a continuous analog of the gradient method

In a real Hilbert space H we consider the nonlinear operator equation P(x)=0 and the continuous gradient methodx (t)= –P (x)^* P (x), x (0) = x₀. Two theorems on the convergence of the process (*) to the solution of the equation P(x)=0 are proved.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 421–426, April, 1968.     

Force enhancement and stability of finite element muscle models

A continuum-mechanical finite element model of skeletal muscle contractions that includes force enhancement based on actin-titin interaction is presented. This model can simulate muscles with a descending limb in the total force-length relation, which has previously led to unstable behaviour. The model predictions are in agreement with results of active stretch experiments. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Rosen's gradient projection methods

This paper is a survey of Rosen's projection methods in nonlinear programming. Through the discussion of previous works, we propose some interesting questions for further research, and also present some new results about the questions.This work was supported in part by the National Science Foundation of China.     

On optimal step-length gradient eigensolvers

Gradient iterations for the minimization of the Rayleigh quotient are robust and (with a proper preconditioning) fast iterations to compute approximations of the smallest eigenvalue of a self-adjoint elliptic partial differential operator. Up to now sharp convergence estimates were only known for the basic fixed-step size preconditioned gradient iteration (also called preconditioned inverse iteration). Recently sharp convergence estimates have been proved for optimal step size (preconditioned) gradient iterations. These new estimates are compared with previous results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On separation of gradient Young measures

Let be a linear subspace of real matrices without rank-one matrices and let be a finite set. Suppose is a bounded arcwise connected Lipschitz domain and is a sequence of bounded vector-valued mappings in such that in as , where is the closed -neighbourhood and the distance function to . We give estimates for such that up to a subsequence, in for some fixed . In other words, we give estimates on such that separates gradient Young measure. The two point set with is a special case of such sets up to a translation. Received: 20 July 2001 / Accepted: 23 January 2002 / Published online: 5 September 2002     

On the complexity of the gradient of a rational function

The Baur-Strassen method implies L(?f) ? 4L(f), where L(f) is the complexity of computing a rational function f by arithmetic circuits, and ?f is the gradient of f. We show that L(? f) ? 3L(f) + n, where n is the number of variables in f. In addition, the depth of a circuit for the gradient is estimated.