International Journal for Numerical Methods in Engineering最新文献

A backward stable numerical calculation of a function with condition number $��\kappa �$ will have a relative accuracy of $��\kappa �{�ϵ�}_{�\text{machine}�}�$. Standard formulations and software implementations of finite-strain elastic materials models make use of the deformation gradient $��F�=�I�+�\partial �u�/�\partial �X�$ and Cauchy-Green tensors. These formulations are not numerically stable, leading to loss of several digits of accuracy when used in the small strain regime, and often precluding the use of single precision floating point arithmetic. We trace the source of this instability to specific points of numerical cancellation, interpretable as ill-conditioned steps. We show how to compute various strain measures in a stable way and how to transform common constitutive models to their stable representations, formulated in either initial or current configuration. The stable formulations all provide accuracy of order $��{�ϵ�}_{�\text{machine}�}�$. In many cases, the stable formulations have elegant representations in terms of appropriate strain measures and offer geometric intuition that is lacking in their standard representation. We show that algorithmic differentiation can stably compute stresses so long as the strain energy is expressed stably, and give principles for stable computation that can be applied to inelastic materials.

Imposing nonperiodic boundary conditions for unit cell analyses may be necessary for a number of reasons in applications, for example, for validation purposes and specific computational setups. The work at hand discusses a strategy for utilizing the powerful technology behind fast Fourier transform (FFT)-based computational micromechanics—initially developed with periodic boundary conditions in mind—for essential boundary conditions in mechanics, as well, for the case of the discretization on a rotated staggered grid. Introduced by F. Willot into the community, the rotated staggered grid is presumably the most popular discretization, and was shown to be equivalent to underintegrated trilinear hexahedral elements. We leverage insights from previous work on the Moulinec–Suquet discretization, exploiting a finite-strain preconditioner for small-strain problems and utilize specific discrete sine and cosine transforms. We demonstrate the computational performance of the novel scheme by dedicated numerical experiments and compare displacement-based methods to implementations on the deformation gradient.

An adjoint-based procedure to determine weaknesses, or, more generally, the material properties of structures is developed and tested. Given a series of load cases and corresponding displacement/strain measurements, the material properties are obtained by minimizing the weighted differences between the measured and computed values. The present paper proposes and tests techniques to minimize the number of load cases and sensors. Several examples show the viability, accuracy and efficiency of the proposed methodology and its potential use for high fidelity digital twins.

We propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non-iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w-invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non-proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial-boundary value problem is solved.

While direct homogenisation approaches such as the FE$��{�\hspace{0.17em}�}^{�2�}�$ method are subject to the assumption of scale separation, the mesh-in-element (MIEL) approach is based on an approach with strong scale coupling, which is based on a discretization with finite elements. In this contribution we propose a two-scale MIEL scheme in the framework of the theory of porous media (TPM). This work is a further development of the MIEL method which is based on the works of the authors A. Ibrahimbegovic, R.L. Taylor, D. Markovic, H.G. Matthies, R. Niekamp (in alphabetical order); where we find the physical and mathematical as well as the software coupling implementation aspects of the multi-scale modeling of heterogeneous structures with inelastic constitutive behaviour, see for example, [Eng Comput, 2005;22(5-6):664-683.] and [Eng Comput, 2009;26(1/2):6-28.]. Within the scope of this contribution, the necessary theoretical foundations of TPM are provided and the special features of the algorithmic implementation in the context of the MIEL method are worked out. Their fusion is investigated in representative numerical examples to evaluate the characteristics of this approach and to determine its range of application.

In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE$��{\hspace{0.17em}}^{�2�}�$ and IGA$��{\hspace{0.17em}}^{�2�}�$ approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null-space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE$��{\hspace{0.17em}}^{�2�}�$ and IGA$��{\hspace{0.17em}}^{�2�}�$ formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems.

This work studies the impact between spherical rigid bodies in the frame of nonsmooth contact dynamics considering friction effects. A new impact element formulation based on the classical instantaneous local Newton impact law is presented. The kinematics properties of the spheres are described by a rigid body formulation with translational and rotational degrees of freedom referred to an inertial frame. In addition, an extension of the nonsmooth generalized-$��\alpha �$ time integration scheme applied to collisions with multiple impacts including Coulomb's friction law is given. Six numerical examples are presented to evaluate the robustness and the performance of the proposed methodology.

Failure probability function (FPF) can reflect quantitative effects of random input distribution parameter (DP) on failure probability, and it is significant for decoupling reliability-based design optimization (RBDO). But the FPF estimation is time-consuming since it generally requires repeated reliability analyses at different DPs. For efficiently estimating FPF, an augmented directional sampling (A-DS) is proposed in this paper. By using the property that the limit state surface (LSS) in physical input space is independent of DP, the A-DS establishes transformation of LSS samples in standard normal spaces corresponding to different DPs. By the established transformation in different standard normal spaces, the LSS samples obtained by DS at a given DP can be transformed to those at other DPs. After simple interpolation post-processing on those transformed samples, the failure probability at other DPs can be estimated by DS simultaneously. The main novelty of A-DS is that a strategy of sharing DS samples is designed for estimating the failure probability at different DPs. The A-DS avoids repeated reliability analyses and inherits merit of DS suitable for solving problems with multiple failure modes and small failure probability. Compared with other FPF estimation methods, the examples sufficiently verify the accuracy and efficiency of A-DS.

Sliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high-speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential $��A^{\prime }�$ and scalar potential $��{\varphi }^{\prime }�$ taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher-order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two-dimensional railgun model and analyzing the current density distribution in a three-dimensional railgun model.

The moving morphable components (MMC) method has been widely used for topology optimization due to its ability to provide an explicit description of topology. However, the MMC method may encounter the instability issue during iteration. Specifically, the iteration history is highly sensitive to parameters of the optimizer, that is, the move limits in the method of moving asymptotes (MMA). Additionally, the final topology obtained from the MMC method usually depends on the initial values. To address these issues and improve the stability of the MMC method in practical applications, this article introduces two strategies. The first strategy is based on the time-series MMC (TSMMC) method, which proposes a unified description of curved components. However, the use of control-points-based design variables may introduce instability into the iteration process due to the strong locality associated with these variables. To mitigate this, global design variables have been incorporated into the formulation. Numerical examples demonstrate that this mixed formulation, combining global and local design variables, can enhance stability significantly. To further enhance stability, the second strategy involves using the trust region-based moving asymptotes (TRMA) method as the optimizer instead of MMA. The TRMA method incorporates an accuracy control mechanism, resulting in stable and fast convergence behavior, as demonstrated in the numerical examples.