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This paper is the third of a series where the convergence analysis of SPH method for multidimensional conservation laws is analyzed. In this paper, two original numerical models for the treatment of boundary conditions are elaborated. To take into account nonlinear effects in agreement with Bardos, LeRoux and Nedelec boundary conditions ([1], [14]), the state at the boundary is computed by solving appropriate Riemann problems. The first numerical model is developed around the idea of boundary forces in surrounding walls, recently initiated in [33] by Monaghan in his simulation of gravity currents. The second one extends the well-known approach of ghost particles for plane boundaries to the case of general curved boundaries. The convergence analysis in L p loc (p < ∞) is achieved thanks to the uniqueness result of measure-valued solutions recently established in [3] for L ∞ initial and boundary data.

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A numerical investigation of the effect of a sudden rupture of a cable component in a Tensegrity assemblage is performed using nonlinear dynamic time history analysis to determine element axial forces and nodal displacements responses of the system. Details of the modeling of the Tensegrity elements as well as the numerical scheme used to integrate the equations of motion are discussed. The investigation is carried on a continuous struts Tensegrity system. The system is constituted by an assemblage of five expanded-octahedron basic modules. The simulation of the sudden rupture of a single cable is realized by replacing this element by its tension, applied as an external force to the extremities of the struts to which this element is attached. This force is then removed instantaneously provoking a load imbalance in the system. This imbalance acts as a triggering mechanism sending an impulsive shock to the damaged structure resulting in a damped vibration motion and large magnitude element axial forces.

Abstract. Uniqueness and existence of L ∞ solutions to initial boundary value problems for scalar conservation laws, with continuous flux functions, is derived by L 1 contraction of Young measure solutions. The classical Kruzkov entropies, extended in Bardos, LeRoux and Nedelec's sense to boundary value problems, are sufficient for the contraction. The uniqueness proof uses the essence of Kruzkov's idea with his symmetric entropy and entropy flux functions, but the usual doubling of variables technique is replaced by the simpler fact that mollified measure solutions are in fact smooth solutions. The mollified measures turn out to have not only weak but also strong boundary entropy flux traces. Another advantage with the Young measure analysis is that the usual assumption of Lipschitz continuous flux functions can be relaxed to continuous fluxes, with little additional work.1. Background to Scalar Conservation Laws with Boundary Conditions. DiPerna [11] showed that measure valued solutions are useful to prove convergence of approximations to scalar conservation laws: convergence follows by verifying that the approximations are uniformly bounded in L ∞ , weakly consistent with all entropy inequalities and consistent with the inititial data, cf. also The present work, with continuous flux functions shows that the Kruzkov entropies, in Bardos, LeRoux and Nedelec's sense, are sufficient for L 1 contraction of Young measure solutions, which in turn implies uniqueness of L ∞ solutions. The uniqueness proof uses the essence of Kruzkov's idea with his symmetric entropy and entropy flux functions, but the usual doubling of variables technique is replaced by the simpler fact that mollified measure solutions are in fact smooth solutions. The mollified measures turn out to also have strong boundary entropy flux traces. Existence and uniqueness for the pure initial value problem, with continuous flux functions, was established by semi group methods in [9] and by measure solutions in [20].At the hart of the matter of initial boundary value problems to scalar conservation laws is the trace of entropy fluxes, which define the boundary condition. The first study [2] used solutions with bounded variation and hence their trace exist directly. The work [18] used the equation in the interior domain to show that the entropy flux, *