Thomas B. J. Di Giusto, Chun Hean Lee, Antonio J. Gil, Javier Bonet, Matteo Giacomini
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In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first-order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex-based Finite Volume method is employed and suitably adapted. 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引用次数: 0
摘要
本文以一阶守恒定律系统的形式,介绍了一种使用新型任意拉格朗日欧拉(ALE)形式的计算框架。除了通常的材料和空间构型外,还引入了额外的参照(内在)构型,以便将材料粒子与网格位置分离。以等温超弹性为起点,编写质量、线性动量和总能量守恒方程,并根据参考构型进行求解。此外,为了保证应变/应力和速度/位移的等阶收敛,标准变形梯度张量(从材料到空间构型的测量)的计算是通过将其乘法分解为两个辅助变形梯度张量而获得的,这两个张量都是通过附加的一阶守恒定律计算的。最重要的是,新的 ALE 保守公式将被证明可以优雅地退化为其他混合守恒定律系统,如总拉格朗日、欧拉和更新参考拉格朗日。将展示守恒定律系统的双曲性,并介绍精确的波速边界,后者对于确保显式时间积分器的稳定性至关重要。在空间离散化方面,采用了基于顶点的有限体积法,并进行了适当调整。为了从连续性和半离散化两个角度保证稳定性,我们精心设计并提出了适当的数值界面通量(通过朗金-胡戈尼奥特跃迁条件)。通过使用系统哈密顿的时间变化来证明稳定性,从而确保数值熵的正向产生。将提出一系列三维基准问题,以证明该框架的稳健性和可靠性。例子将仅限于等温可逆弹性情况,以展示新公式的潜力。
A first-order hyperbolic arbitrary Lagrangian Eulerian conservation formulation for non-linear solid dynamics
The paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first-order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first-order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex-based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi-discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation.
期刊介绍:
The International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems.
The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome.