Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	DoctorJu Liu,Southern University of Science and Technology
Inviter:	Professor Yifa Tang
Title:	Recent advancements in structure-preserving integration for nonlinear and inelastic problems
Language:	Chinese
Time & Venue:	2023.08.07 09:00-10:30 N702
Abstract:	Preserving physical and geometrical structures in dynamic calculations has been demonstrated to be critical in ensuring reliable results. The concept of energy-momentum scheme and the discrete gradient strategy have enabled the design of energy-momentum consistent integrators for shell dynamics, multibody systems, elastoplasticity, viscoelasticity, contact, and impact, to list a few. Abundant numerical evidence has justified its advantage over classical implicit integrators, especially for long-term simulations. In this talk, I will discuss recent advancements in designing structure-preserving integrators. First, I will focus on the discrete gradient formula initially proposed by O. Gonzalez. Its quotient form renders it to be numerically singular when the adjacent solutions are close. Leveraging a set of specially developed quadrature rules, the potential energy can be split into two parts, each of which is treated separately to guarantee a dissipative nature in the numerical residual. The resulting integrators are non-singular and energy- momentum consistent. Interestingly, they are found to be dissipative for the high-frequency modes without harming the relative equilibria. The technique can be utilized to improve the robustness of existing conserving integrators. Second, I will focus on incompressible elastodynamics and propose a new Hamiltonian. Based on a recently proposed mixed formulation and the discrete gradient formula, this Hamiltonian and momenta can be conserved in the fully discrete level. The scaled mid-point formula, another popular option for constructing algorithmic stresses, is analyzed and demonstrated to be non-robust numerically. The generalized Taylor-Hood element based on the spline technology offers a higher-order, robust, and inf-sup stable spatial discretization option. This element technology is further enhanced by the grad-div stabilization to improve its discrete mass conservation. I will also discuss and evaluate the spectral decomposition algorithms for stretch-based models. Third, I will discuss a viscoelastic model that gained popularity over the years but was recently identified to suffer from thermodynamic inconsistency. I will revisit this model by proposing a complete thermomechanical theory of it. The derivation elucidates the origin of the evolution equations of that model, with a few non-negligible differences. Based on the consistent framework, an energy-momentum scheme is constructed using a strain-driven approach and a generalized directionality property for the stress-like variable. Numerical examples will be provided to justify the effectiveness of the overall methodology. I will draw conclusions and discuss some promising directions at the end of this talk.