Sensitivity Analysis for the Stationary Distribution of Reflected Brownian Motion in a Convex Polyhedral Cone

Reflected Brownian motion (RBM) in a convex polyhedral cone arises in a variety of applications ranging from the theory of stochastic networks to mathematical finance, and under general stability conditions, it has a unique stationary distribution. In such applications, to implement a stochastic optimization algorithm or quantify robustness of a model, it is useful to characterize the dependence of stationary performance measures on model parameters. In this paper, we characterize parametric sensitivities of the stationary distribution of an RBM in a simple convex polyhedral cone, that is, sensitivities to perturbations of the parameters that define the RBM—namely the covariance matrix, drift vector, and directions of reflection along the boundary of the polyhedral cone. In order to characterize these sensitivities, we study the long-time behavior of the joint process consisting of an RBM along with its so-called derivative process, which characterizes pathwise derivatives of RBMs on finite time intervals. We show that the joint process is positive recurrent and has a unique stationary distribution and that parametric sensitivities of the stationary distribution of an RBM can be expressed in terms of the stationary distribution of the joint process. This can be thought of as establishing an interchange of the differential operator and the limit in time. The analysis of ergodicity of the joint process is significantly more complicated than that of the RBM because of its degeneracy and the fact that the derivative process exhibits jumps that are modulated by the RBM. The proofs of our results rely on path properties of coupled RBMs and contraction properties related to the geometry of the polyhedral cone and directions of reflection along the boundary. Our results are potentially useful for developing efficient numerical algorithms for computing sensitivities of functionals of stationary RBMs.

Multisphere Representation of Convex Polyhedral Particles for DEM Simulation

The representation of particles of complex shapes is one of the key challenges of numerical simulations based on the discrete element method (DEM). A novel algorithm has been developed by the authors to accurately represent 2D arbitrary particles for DEM modelling. In this paper, the algorithm is extended from 2D to 3D to model convex polyhedral particles based on multisphere methods, which includes three steps: the placement of spheres at the corners, along the edges, and on the facets in sequence. To give a good representation of a polyhedral particle, the spheres are placed tangent to the particle surface in each step. All spheres placed in the three steps are clumped together into a clump in DEM. In addition, the mass properties of the clump are determined based on the corresponding polyhedral particle to obtain accurate simulation results. Finally, an example is used to validate the robust and automatic performance of the algorithm in generating a sphere clump model for an assembly of polyhedral particles. A current FORTRAN version of the algorithm is available by contacting the authors.

Three-Dimension Random Aggregate Generation of Numerical Concrete Model

The aggregate generation of concrete is one of the important problems in concrete mesoscopic mechanics. Firstly, the mesoscopic numerical model with spherical aggregates is obtained by the method of excluding the occupied space, and fully-graded concrete model of high aggregate content can be quickly generated. Then, based on the spherical aggregate model, the generation method of random convex polyhedral aggregates is proposed. Finally, a full-graded concrete model with spherical aggregates is shown in Case 1 and a cylindrical concrete model with random convex polyhedral aggregates is shown in Case 2. The result shows that the aggregates are equally distributed in the concrete models which can be used in the study of mesoscopic numerical calculation.