Vol 59, No 12 (2019)

Unlike other schemes that locally violate the essential stability properties of the analytic parabolic and elliptic problems, Voronoi finite volume methods (FVM) and boundary conforming Delaunay meshes provide good approximation of the geometry of a problem and are able to preserve the essential qualitative properties of the solution for any given resolution in space and time as well as changes in time scales of multiple orders of magnitude. This work provides a brief description of the essential and useful properties of the Voronoi FVM, application examples, and a motivation why Voronoi FVM deserve to be used more often in practice than they are currently.

An ingenious construction of Gel’fand, Kapranov, and Zelevinsky [5] geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with nonempty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a nondegenerated coherent triangulation to another nondegenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are nondegenerated and coherent. As an application of this theory, we propose an algorithm to triangulate a special class of 3d nonconvex polyhedra without using additional vertices. We prove that this algorithm terminates in \(O({{n}^{3}})\) time.

The majority of problems in structural computational biology require minimization of the energy function (force field) defined on the molecule geometry. This makes it possible to determine properties of molecules, predict the correct arrangement of protein chains, find the best molecular docking for complex formation, verify hypotheses concerning the protein design, and solve other problems arising in modern drug development. In the case of low-molecular compounds (consisting of less than 250 atoms), the problem of finding the geometry that minimizes the energy function is well studied. The minimization of macromolecules (in particular, proteins) consisting of tens of thousands of atoms is more difficult. However, a distinctive feature of statements of these problems is that initial approximations that are close to the solution are often available. Therefore, the original problem can be formulated as a problem of nonconvex optimization in the space of about \({{10}^{4}}\) variables. The complexity of computing the function and its gradient is quadratic in the number variables. A comparative analysis of gradient-free methods with gradient-type methods (gradient descent, fast gradient descent, conjugate gradient, and quasi-Newton methods) in their GPU implementations is carried out.

A Newton-type method is proposed for numerical minimization of convex piecewise quadratic functions, and its convergence is analyzed. Previously, a similar method was successfully applied to optimization problems arising in mesh generation. It is shown that the method is applicable to computing the projection of a given point onto the set of nonnegative solutions of a system of linear equations and to determining the distance between two convex polyhedra. The performance of the method is tested on a set of problems from the NETLIB repository.

A new method is proposed for approximating the Edgeworth–Pareto hull of a feasible objective set in a multiobjective optimization (MOO) problem with criteria functions having numerous local extrema. The method is based on constructing a launch pad, i.e., a subset of the feasible decision set such that gradient procedures for local optimization of criteria and scalarizing functions of criteria starting at these points yield efficient decisions of the MOO problem. A launch pad is constructed using the optima injection method, which combines the usual multistart approach with a genetic algorithm for Pareto frontier approximation. It is shown that the proposed launch pad method (LPM) can also be used to approximate the effective hull of a nonconvex multidimensional set. A theoretical analysis of LPM is presented, and experimental results are given for the applied problem of constructing control rules for a cascade of reservoirs, which is reduced to a complicated MOO problem with scalarizing functions having numerous local extrema.

In this paper, three-dimensional mesh curving and refinement methods are examined for high-order flow simulations with discontinuous Galerkin (DG) methods on hybrid grids. The mesh curving algorithm converts linear surface elements to quadratic ones with the cubic Bézier surface reconstruction. The effects of mesh curving on the impacts of DG solutions of the Euler and Navier–Stokes equations are investigated. Numerical results show that significant enhancements of accuracy and robustness can be gained for DG solutions of smooth and discontinuous flow fields. Additionally, a curved mesh refinement algorithm is also realized by inquiring the midpoints of edges and faces of the reconstructed quadratic elements. With this method, up to 0.9 billons curved elements are successfully generated around the DLR-F6 wing/body/nacelle/pylon configuration.

In high performance computing, block structured grids are favored due to their geometric adaptability while supporting computational performance optimizations connected with structured grid discretizations. However, many problems on geometrically complex domains are traditionally solved using fully unstructured (usually simplicial) meshes. We attempt to address this deficiency in the two-dimensional case by presenting a method which generates block structured grids with a prescribed number of blocks from an arbitrary triangular grid. Special attention was paid to mesh quality while simultaneously allowing for complex domains. Our method guarantees fulfillment of user-defined minimal element quality criteria—an essential feature for grid generators in simulations using finite element or finite volume methods. The performance of the proposed method is evaluated on grids constructed for regional ocean problems utilizing two-dimensional shallow water equations.

For an elliptic equation in divergent form with a discontinuous scalar piecewise constant coefficient in an unbounded domain \(\Omega \subset {{\mathbb{R}}^{2}}\) with a piecewise smooth noncompact boundary and smooth discontinuity lines of the coefficient, the \({{L}_{p}}\)-interaction of a finite and an infinite singular points of a weak solution to the Dirichlet problem is studied in a class of functions with the first derivatives from \({{L}_{p}}(\Omega )\) in the entire range of the exponent \(p \in (1,\infty )\).