Numerical differentiation for local and global tangent operators in computational plasticity

Abstract

In this paper, numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators. The derivatives of the constitutive equations are approximated by means of difference schemes. These derivatives are needed to achieve quadratic convergence in the integration at Gauss-point level and in the solution of the boundary value problem. Numerical differentiation is shown to be a simple, robust and competitive alternative to analytical derivatives. Quadratic convergence is maintained, provided that adequate schemes and stepsizes are chosen. This point is illustrated by means of some numerical examples.

Consistent tangent matrices for substepping schemes

“…The approximated derivatives are then used to solve the local problems and to compute the consistent tangent matrix. As shown in References [7] and [8], this approach is both robust and computationally efficient. Quadratic convergence is maintained, provided that adequate difference schemes and stepsizes (i.e.…”

“…For the MRS-Lade model, not all the derivatives needed to compute the Jacobian of the residual, equation (4), have a readily available analytical expression. Because of this, and following References [7] and [8], numerical differentiation is used to approximate the Jacobian. In these references, numerical differentiation is combined with the backward Euler integration rule and quadratic convergence is obtained at local and global level.…”

“…Going beyond References [7] and [8], substepping and numerical differentia-tion are combined here. Like in the first example, the adaptive substepping technique is activated at the Gauss points where the local problem requires more than 6 iterations for convergence (up to a tolerance of 10 −12 ).…”

“…Going beyond References [7] and [8], where only the single-step backward Euler rule is considered, this paper shows that substepping and numerical differentiation can be combined to obtain the consistent tangent matrix leading to quadratic convergence in complex situations, where 1) analytical derivatives of the constitutive equation are not available and 2) the use of single-step integration rules would demand prohibitively small time-steps. This point is illustrated in some examples with the MRS-Lade model [8,9,10].…”

Field Assisted Sintering Technology, Part II: Simulation

“…Here, several possibilities are available for the computation of this derivative. Either analytical derivations, numerical differentiation, see [16][17][18], or automatic differentiation are possible, see, for example, [19][20][21]. More modern aspects, which are based on optimality conditions, are proposed, for example, in [22].…”