Dual-support smoothed particle hydrodynamics for elastic mechanics Zili Dai 3 , Huilong Ren 3 , Xiaoying Zhuang 3,4,∗ , Timon Rabczuk 1,2,3,∗ 1 Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam 2 Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam 3 Institute of Structural Mechanics, Bauhaus-Universit at Weimar, Germany 4 Department of Geotechnical Engineering, College of Civil Engineering,Tongji University, China Abstract In the standard SPH method, the interaction between two particles might be not pairwise when the support domain varies, which can result in a reduction of accuracy. To deal with this problem, a modified SPH approach is presented in this paper. First of all, a Lagrangian kernel is introduced to eliminate spurious distortions of the domain of material stability, and the gradient is corrected by a linear transformation so that linear completeness is satisfied. Then, concepts of support and dual-support are defined to deal with the unbalanced interactions between the particles with different support domains. Several benchmark problems in one, two and three dimensions are tested to verify the accuracy of the modified SPH model and highlight its advantages over the standard SPH method through comparisons. Keywords : SPH, dual-support, elastic mechanics, unbalanced interaction, wave reflection 1 Introduction Smoothed particle hydrodynamics (SPH) is a mesh-free technique based on a pure La- grangian description. Originally developed for astrophysical applications by Lucy [35] and Gingold and Monaghan [19], it has been widely adapted by a range of problems in various disciplines [8, 10, 11, 20, 22, 29, 32, 31, 38, 43, 53, 52]. As a meshless technique, the main advantage of SPH method is to bypass the need of a numerical grid to calculate spatial derivatives. Hence, it avoids the problems associated with mesh tangling and dis- tortion, which usually occur in FEM analyses for large deformation impact and explosive loading events. As a Lagrangian technique, it offers advantages in problems with moving boundaries, large deformations, dynamic fracture and multiple phases. The first SPH application to problems in the framework of solid mechanics was con- ducted by Libersky and Petschek [26]. Since then, there has been a growing interest in applying SPH to a wide variety of solid mechanics problems with many promising results [7, 9, 18, 21, 30, 41, 43, 44, 45, 61, 62]. While the favorable features of the SPH method and its applications to solid mechanics have been noted, drawbacks, such as inconsistency [3, 40] , tensile instability [21, 57], and zero energy modes [56], have also been identified. Accordingly, various remedies were proposed for these problems and improved the com- puting accuracy and stability [6, 15, 27, 42, 48, 58]. In SPH, the field variables of each particle are estimated as sums over all neighboring particles which are located in its support domain. In many problems involving plastic de- formation or damage, high stress concentrations occur which require a finer discretization with smaller (or varying) support size. When the sizes of support domain change, the interaction of the particles might be unilateral, thus resulting in an unbalanced internal 1 arXiv:1703.07209v1 [physics.comp-ph] 21 Mar 2017