BDF-like methods for nonlinear dynamic analysis
Introduction
The failure of popular algorithms in nonlinear dynamic analysis, e.g. the loss of unconditional stability of the trapezoidal rule in the nonlinear regime [32], [40], [26], [1], has motivated much of recent work in the development of more robust time integration algorithms for nonlinear elastodynamics. As pointed out in [25], numerical stability is of primary importance when developing such schemes. In this regard, energy-conserving algorithms (e.g. [22], [36], [26], [23], [17], [8], [34], [29], [33]) that target nonlinear problems have been proposed by a number of researchers. Among them the energy-momentum type methods pioneered by Simo and Tarnow [36], with improvements and extensions by many subsequent efforts (see e.g. [37], [26], [25], [17], [28], among others), have been especially successful. However, energy-conserving schemes have shown difficulties for numerically stiff problems due to their lack of dissipation in the high-frequency range. Failures of energy-conserving algorithms have been reported in [31], [6], [26], [27], [2], among others. It has been realized that the numerical instabilities associated with the existence of repeated unit root at infinite frequency in common conserving schemes result in highly oscillatory responses, which hinder the convergence process for the solution of nonlinear equations [2]. Reducing the time step size may not necessarily help the convergence process as a smaller time step may allow the excitation of even higher frequencies [5]. As a result, the need for numerical dissipation in the high-frequency range, even though the underlying system may exhibit full energy conservation, has been commonly recognized for robust time integration algorithms in the nonlinear regime.
Classical dissipative schemes [30], [38], [20], [39], [9] have been developed in the context of linear elastodynamics, see [21] for a more comprehensive description. Although they have also been applied to nonlinear problems, it is observed that these algorithms fail to provide reliable high-frequency dissipation in the nonlinear regime [25], [2]. Indeed, the value of the algorithmic parameter at which the scheme is dissipative may become problem dependent, see [19] for such an example in the nonlinear regime with the HHT- scheme [20]. Much of recent research work [26], [25], [7], [2], [5], [19] aiming to provide reliable numerical dissipation in the nonlinear regime has been motivated by the ineffectiveness of classical dissipative schemes for nonlinear problems. Most of the recently proposed algorithms are constructed based on some energy-conserving scheme, or other approaches such as the time discontinuous Galerkin method.
Another interesting approach has recently been proposed by Bathe and collaborators [3], [4]. The main idea is to combine the trapezoidal rule and the second-order backward Euler method into a composite algorithm. High-frequency numerical dissipation is introduced through the backward Euler component. The algorithm has been demonstrated to be effective for nonlinear elastodynamic problems involving large deformations, where the trapezoidal rule fails to produce a stable solution. The simplicity of this approach is particularly noteworthy, together with the symmetry of the resultant tangential stiffness matrix, which is to be contrasted with the non-symmetry of the tangent matrices resulting from, for example, the energy-momentum based methods.
In this paper we propose a general four-step scheme that bears a resemblance to the backward differentiation formulas (BDF) [16], and present two time integration algorithms based on this scheme. We also consider a composite algorithm incorporating such a BDF-like scheme and the trapezoidal rule using a composite strategy similar to the Bathe method [3], and also present an extension of the Bathe composite strategy. These algorithms each involve two algorithmic parameters. The domains of the appropriate parameter values are determined based on a linear stability analysis and the consideration of dissipativity. Although a nonlinear stability analysis of these algorithms for general nonlinear elastodynamic problems is still elusive, numerical experiments suggest that these algorithms are very effective for nonlinear dynamic problems at time step sizes where the trapezoidal rule encounters a well-known instability. We test these algorithms for several three-dimensional (3D) nonlinear elastodynamic problems involving large deformations with St. Venant-Kirchhoff and compressible Neo-Hookean material models. The convergence characteristics of these algorithms are demonstrated using a nonlinear problem having analytic solutions.
The rest of this paper is organized as follows. In Section 2 we briefly discuss the high-order spatial discretization scheme of the nonlinear elastodynamic equation with the spectral element method, which has been documented in detail elsewhere [15]. The proposed temporal algorithms will be implemented and tested in conjunction with this approach for spatial discretization. In Section 3 we present a general four-step BDF-like scheme with second-order accuracy, and several algorithms based on this scheme. In Section 4 we demonstrate the temporal convergence characteristics of these algorithms with a nonlinear problem having analytic solutions. In Section 5 we test the proposed algorithms with several nonlinear elastodynamic problems involving large deformations for St. Venant-Kirchhoff and Neo-Hookean hyperelastic materials, and compare them with the trapezoidal rule, the Bathe method, and the Park method [32]. Finally, Section 6 provides some concluding remarks.
Section snippets
Problem formulation
Consider the finite deformation of a 3D object occupying domain with boundary , where Dirichlet boundary conditions (BC) are provided on and Neumann-type (traction) BCs on . Assume that the object is in its natural configuration (no deformation), , at time t = 0, and deforms to a new configuration, , at time t. With respect to the initial configuration , the weak form of the momentum equation can be expressed as follows,
Time integration algorithms
After spatial discretization of Eq. (1), we obtain a semi-discretized equation,where overdot denotes the time derivative; M and U are, respectively the mass matrix and the vector of expansion coefficients of the displacement; N represents the contribution of the internal stresses, and is nonlinear with respect to the displacement; R represents the contribution of the external loads. Note that R does not depend on U under the assumption of non-follower loads we made in Section 2
Convergence characteristics
The goal of this section is to numerically demonstrate the convergence characteristics of the time integration schemes presented in the previous section. We consider the nonlinear vibration, with finite deformation throughout time, of a cubic object of a compressible Neo-Hookean material, whose motion is described by an analytic solution to the nonlinear elastodynamic equation.
Specifically, we consider the cubic block depicted in Fig. 10, which initially occupies the domain
Representative numerical examples
To evaluate the performance of the proposed time integration algorithms, we consider several three-dimensional numerical example problems of nonlinear elastodynamics. We solve these problems with the proposed algorithms in Section 3, and compare the results with those from the trapezoidal rule, the Bathe method [3], and the Park method [32]. The test problems involve large deformations, large displacements and rotations, which demonstrate the difficulties encountered by some algorithms. The
Concluding remarks
In this study we have presented several second-order time integration algorithms based on a BDF-like scheme, together with an extension of the Bathe composite strategy. The domains of appropriate algorithmic parameter values have been determined through a linear stability analysis, and less dissipative members of each algorithm have been identified. We have tested these algorithms with several three-dimensional nonlinear elastodynamic problems. The material models in these tests include St.
Acknowledgment
The author gratefully acknowledges the support from the NSF and the DOE/PSAAP program. Computer time was provided by the TeraGrid through an MRAC grant and by the Rosen Center for Advanced Computing at Purdue University.
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