# Methods and Applications of Analysis

## Volume 28 (2021)

### Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part II

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

### A fully compatible staggered Lagrangian algorithm for elastic-plastic flows utilizing the conservation of total energy

Pages: 355 – 370

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n3.a7

#### Authors

Yueling Jia (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Song Jiang (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Jun-Bo Cheng (Institute of Applied Physics and Computational Mathematics, Beijing, China)

#### Abstract

In this paper we construct a fully compatible staggered Lagrangian algorithm for the equations of two-dimensional elastic-plastic flows (FCSLAEP), with the hypo-elastic incremental constitutive model, von Mises’ yielding condition and the Mie–Grüneisen equation of state. To construct our scheme, we first reformulate all governing equations of elastic-plastic flows, including the equations of deviatoric stress, into a hyperbolic system in the form of the divergence and the gradient operators. Then, this hyperbolic system is discretized by adapting the method of support operators and using some new vector identities of differential calculus. Moreover, we replace the finite volume surface integrals with the line integrals and rewrite the equations of deviatoric stress in the form of the internal energy on the right-hand side (so that one can use the gradient operator on the velocity in the discrete form), to discretize the equations of deviatoric stress in order to conserve the total energy and preserve the symmetry. Finally, the predictor-corrector technique is used with respect to time to improve the accuracy. A number of numerical tests are carried out, and the numerical results show that the proposed scheme FCSLAEP seems robust and convergent. Moreover, the scheme is of the 2nd order accuracy, and conserves the total energy and preserves the symmetry.

#### Keywords

2D elastic-plastic flows, hypo-elastic constitutive model, Mie–Grüneisen equation of state, fully compatible staggered Lagrangian algorithm, conservation of total energy, preservation of symmetry

#### 2010 Mathematics Subject Classification

68U05, 74C05, 74F10, 74M20

The research of Jiang was supported by the National Key R&D Program (2020YFA0712200), the National Key Project (GJXM92579), NSFC (Grant No. 11631008, No. 12072043), the Sino-German Science Center (Grant No. GZ 1465), and the ISF-NSFC joint research program (Grant No. 11761141008).
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