Abstract
We present the application of Harten-Lax-van Leer (HLL)-type solvers on Riemann problems in nonlinear elasticity which undergoes high-load conditions. In particular, the HLLD (“D” denotes Discontinuities) Riemann solver is proved to have better robustness and efficiency for resolving complex nonlinear wave structures compared with the HLL and HLLC (“C” denotes Contact) solvers, especially in the shock-tube problem including more than five waves. Also, Godunov finite volume scheme is extended to higher order of accuracy by means of piecewise parabolic method (PPM), which could be used with HLL-type solvers and employed to construct the fluxes. Moreover, in the case of multi material components, level set algorithm is applied to track the interface between different materials, while the interaction of interfaces is realized through HLLD Riemann solver combined with modified ghost method. As seen from the results of both the solid/solid “stick” problem with the same material at the two sides of contact interface and the solid/solid “slip” problem with different materials at the two sides, this scheme composed of HLLD solver, PPM and level set algorithm can capture the material interface effectively and suppress spurious oscillations therein significantly.
Highlights
Nonlinear elastic deformation of solid material undergoing high-load conditions commonly occurs in industrial application areas, such as the design of automobile anti-collision device, the evaluation of the capability of spacecraft structural materials against hypervelocity impact, and etc
We first present a comparative study of the Harten-Lax-van Leer (HLL)-type Riemann solvers which are applied in single material cases. With the state equation given in Equation (10) employed, Equation (8) is solved in each test case under different initial and boundary condition
The existing Godunov-type shock-capturing schemes have been applied in conjunction with HLL family of Riemann solver to solve Riemann problems in nonlinear elasticity
Summary
Nonlinear elastic deformation of solid material undergoing high-load conditions commonly occurs in industrial application areas, such as the design of automobile anti-collision device, the evaluation of the capability of spacecraft structural materials against hypervelocity impact, and etc. The second one is constructed on the basis of inverse deformation gradient tensor (f = F−1 = ∂X/∂x, namely gradient of Lagrangian coordinates to Eulerian coordinates) This model, which needs to solve 21 equations in three dimensions, was firstly suggested by Plohr & Sharp[3] for elastic solids and developed to be applicable to rate-dependent and rate-independent plasticities. Numerous works are devoted to the improvement of the Godunov scheme, including increasing the precision and resolution of the scheme as well as decreasing the computational time by using the approximate solution of Riemann problem to replace the exact solution (see Garaizar[18], Miller[19], Titarev[20] and Barton[21]) For the former, several researchers derived the scheme with second-order accuracy in space (e.g. MUSCL) by modifying the constant approximation of original variables to linear distribution[4,22]; while for the latter, Titarev[20] discussed several approximate solution methods for nonlinear elasticity, including GMUSTA and EVILIN solvers, linearized method, and FORCE flux method. The highly efficient and reliable numerical method is needed to be developed with interface tracking method to achieve the two objects of studying the interaction between different materials and decreasing numerical oscillation at material interfaces at the same time
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