Non-convex Bodies Research Articles

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by nonconvex immersed bodies, multiply connected domains, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the immersed solid bodies, reducing computational cost and enabling simulations with nonconvex bodies that could not be discretized with previous immersed interface methods. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, allowing for simulations with multiply connected domains and outflow boundaries that have challenged other immersed interface vortex methods. We also explore novel methods for recovering time-dependent pressure distributions on immersed bodies within a vorticity-based method and present a novel control volume formulation for recovering aerodynamic moments from only the vorticity and velocity fields. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains.

Polynomials associated to non-convex bodies

Polynomial spaces associated to a convex body C in \((\mathbb{R}^{+})^d\) have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex C. We develop some basic pluripotential theory including notions of C−extremal plurisubharmonic functions VC,K for \(K\subset \mathbb{C}^d\) compact. Using this, we discuss Bernstein−Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.

On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

Spatial Models of Sugars and Their Compounds

The images of saccharide and polysaccharide molecules in spaces of various dimensions are considered. A method has been developed for obtaining simplified three-dimensional images of sugar molecules and their chains based on their images in spaces of higher dimensions. It was found that three-dimensional images of furanose and pyranose molecules fundamentally differ from each other to form convex and, accordingly, non-convex bodies. This leads to fundamental differences in the structure of polysaccharides from these molecules.

Quantitative characterizations of nonconvex bodies with smooth boundaries in Hilbert spaces via the metric projection

The present paper is a continuation of a previous work with conditions for regularity of the metric projection. Here we provide a full quantitative characterization of closed bodies in Hilbert spaces with Cp+1-smooth boundary in terms of the smoothness of the metric projection and the behavior of its first derivative.

Continuous smoothed particle hydrodynamics for cracked nonconvex bodies by diffraction criterion

The continuous smoothed particle hydrodynamics (CSPH) method is proposed to modify the kernel approximation in the smoothed particle hydrodynamics (SPH). The diffraction criterion is embedded in the framework of standard SPH. CSPH dramatically improves the computational ability in cracked bodies. Three numerical cases of cracked bodies, including dynamic stress intensity factor (DSIF) of a central crack, tensile fracture of brittle materials and Kalthoff’s experiment, are employed to verify the ability of the proposed method. It is found that the numerical results obtained by CSPH are in good agreement with the previous experimental and numerical results, implying the robustness and feasibility of the proposed method.

Newton’s aerodynamic for non-convex bodies

We characterize the solution to the Newton minimal resistance problem in a specific class of hollow profiles, satisfying a q -concavity condition. We treat two-dimensional bodies and radially symmetric three-dimensional bodies.

A parallel Discrete Element Method to model collisions between non-convex particles

In many dry granular and suspension flow configurations, particles can be highly non-spherical. It is now well established in the literature that particle shape affects the flow dynamics or the microstructure of the particles assembly in assorted ways as e.g. compacity of packed bed or heap, dilation under shear, resistance to shear, momentum transfer between translational and angular motions, ability to form arches and block the flow. In this talk, we suggest an accurate and efficient way to model collisions between particles of (almost) arbitrary shape. For that purpose, we develop a Discrete Element Method (DEM) combined with a soft particle contact model. The collision detection algorithm handles contacts between bodies of various shape and size. For nonconvex bodies, our strategy is based on decomposing a non-convex body into a set of convex ones. Therefore, our novel method can be called “glued-convex method” (in the sense clumping convex bodies together), as an extension of the popular “glued-spheres” method, and is implemented in our own granular dynamics code Grains3D. Since the whole problem is solved explicitly, our fully-MPI parallelized code Grains3D exhibits a very high scalability when dynamic load balancing is not required. In particular, simulations on up to a few thousands cores in configurations involving up to a few tens of millions of particles can readily be performed. We apply our enhanced numerical model to (i) the collapse of a granular column made of convex particles and (i) the microstructure of a heap of non-convex particles in a cylindrical reactor.

Smoothness of the metric projection onto nonconvex bodies in Hilbert spaces

Based on a fundamental work of R. B. Holmes from 1973, we study differentiability properties of the metric projection onto prox-regular sets. We show that if the set is a nonconvex body with a Cp+1-smooth boundary, then the projection is Cp-smooth near suitable open truncated normal rays, which are determined only by the function of prox-regularity. A local version of the same result is established as well, namely, when the smoothness of the boundary and the prox-regularity of the set are assumed only near a fixed point. Finally, similar results are derived when the prox-regular set is itself a Cp+1-submanifold.

Self-diffusion in the non-Newtonian regime of shearing liquid crystal model systems based on the Gay-Berne potential.

The self-diffusion coefficients of nematic phases of various model systems consisting of regular convex calamitic and discotic ellipsoids and non-convex bodies such as bent-core molecules and soft ellipsoid strings have been obtained as functions of the shear rate in a shear flow. Then the self-diffusion coefficient is a second rank tensor with three different diagonal components and two off-diagonal components. These coefficients were found to be determined by a combination of two mechanisms, which previously have been found to govern the self-diffusion of shearing isotropic liquids, namely, (i) shear alignment enhancing the diffusion in the direction parallel to the streamlines and hindering the diffusion in the perpendicular directions and (ii) the distortion of the shell structure in the liquid whereby a molecule more readily can escape from a surrounding shell of nearest neighbors, so that the mobility increases in every direction. Thus, the diffusion parallel to the streamlines always increases with the shear rate since these mechanisms cooperate in this direction. In the perpendicular directions, these mechanisms counteract each other so that the behaviour becomes less regular. In the case of the nematic phases of the calamitic and discotic ellipsoids and of the bent core molecules, mechanism (ii) prevails so that the diffusion coefficients increase. However, the diffusion coefficients of the soft ellipsoid strings decrease in the direction of the velocity gradient because the broadsides of these molecules are oriented perpendicularly to this direction due the shear alignment (i). The cross coupling coefficient relating a gradient of tracer particles in the direction of the velocity gradient and their flow in the direction of the streamlines is negative and rather large, whereas the other coupling coefficient relating a gradient in the direction of the streamlines and a flow in the direction of the velocity gradient is very small.

A fast and stable solver for acoustic scattering problems based on the nonuniform grid approach.

A fast and stable boundary element method (BEM) algorithm for solving external problems of acoustic scattering by impenetrable bodies is developed. The method employs the Burton-Miller integral equation, which provides stable convergence of iterative solvers, and a generalized multilevel nonuniform grid (MLNG) algorithm for fast evaluation of field integrals. The MLNG approach is used here for the removal of computational bottlenecks involved with repeated matrix-vector multiplications as well as for the low-order basis function regularization of the hyper-singular integral kernel. The method is used for calculating the fields scattered by large acoustic scatterers, including nonconvex bodies with piece-wise smooth surfaces. As a result, the algorithm is capable of accurately incorporating high-frequency effects such as creeping waves and multiple-edges diffractions. In all cases, stable convergence of the method is observed. High accuracy of the method is demonstrated by comparison with the traditional BEM solution. The computational complexity of the method in terms of both the computation time and storage is estimated in practical computations and shown to be close to the asymptotic O(N log N) dependence.

An impulse-based energy tracking method for collision resolution

Discrete element methods can be based on either penalties or impulses to resolve collisions. A generic impulse based method, the energy tracking method (ETM), is described to resolve collisions between multiple non-convex bodies in three dimensions. As opposed to the standard sequential impulse method (SQM) and simultaneous impulse method (SMM), which also apply impulses to avoid penetration, the energy tracking method changes the relative velocity between two colliding bodies iteratively yet simultaneously. Its main novelty is that impulses are applied gradually at multi-point contacts, and energy changes at the contact points are tracked to ensure conservation. Three main steps are involved in the propagation of the impulses during the single- and multi-contact resolution: compression, restitution-related energy loss, and separation. Numerical tests show that the energy tracking method captures the energy conservation property of perfectly elastic single- and multi-point collisions. ETM exhibits improved angular velocity estimation, as compared to SMM and SQM, as demonstrated by two numerical examples that model multi-point contact between box-shaped objects. Angles of repose estimated for multi-object pack repositioning of spheres, cubes, and crosses are in good agreement with the reported experimental values.

Dense periodic packings of tori.

Dense packings of nonoverlapping bodies in three-dimensional Euclidean space R(3) are useful models of the structure of a variety of many-particle systems that arise in the physical and biological sciences. Here we investigate the packing behavior of congruent ring tori in R(3), which are multiply connected nonconvex bodies of genus 1, as well as horn and spindle tori. Specifically, we analytically construct a family of dense periodic packings of unlinked tori guided by the organizing principles originally devised for simply connected solid bodies [Torquato and Jiao, Phys. Rev. E 86, 011102 (2012)]. We find that the horn tori as well as certain spindle and ring tori can achieve a packing density not only higher than that of spheres (i.e., π/sqrt[18] = 0.7404...) but also higher than the densest known ellipsoid packings (i.e., 0.7707...). In addition, we study dense packings of clusters of pair-linked ring tori (i.e., Hopf links), which can possess much higher densities than corresponding packings consisting of unlinked tori.

Development of contact algorithm for three-dimensional numerical manifold method

SUMMARY This paper customizes a contact detection and enforcing scheme to fit the three-dimensional (3-D) numerical manifold method (NMM). A hierarchical contact system is established for efficient contact detection. The mathematical mesh, a unique component in the NMM, is utilized for global searching of possible contact blocks and elements, followed by the local searching to identify primitive hierarchies. All the potential contact pairs are then transformed into one of the two essential entrance modes: point-to-plane and crossing-lines modes, among which real contact pairs are detected through a unified formula. The penalty method is selected to enforce the contact constraints, and a general contact solution procedure in the 3-D NMM is established. Because of the implicit framework, an open-close iteration is performed within each time step to determine the correct number of contact pairs among multi-bodies and to achieve complete convergence of imposed contact force at corresponding position. The proposed contact algorithm extensively utilizes most of the original components of the NMM, namely, the mathematical mesh/cells and the manifold elements, as well as the external components associated with contacts, such as the contact body, the contact facet and the contact vertex. In particular, the utilization of two mutually approaching mathematical cells is efficient in detecting contacting territory, which makes this method particularly effective for both convex and non-convex bodies. The validity and accuracy of the proposed contact algorithm are verified and demonstrated through three benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd.

A stochastic pseudospectral and T-matrix algorithm for acoustic scattering by a class of multiple particle configurations

We consider absorption and scattering of acoustic waves from uncertain configurations comprising multiple two dimensional bodies with various material properties (sound-soft, sound-hard, absorbing and penetrable) and develop tools to address the problem of quantifying uncertainties in the acoustic cross sections of the configurations. The uncertainty arises because the locations and orientations of the particles in the configurations are described through random variables, and statistical moments of the far-fields induced by the stochastic configurations facilitate quantification of the uncertainty. We develop an efficient algorithm, based on a hybrid of the stochastic pseudospectral discretization (to truncate the infinite dimensional stochastic process) and an efficient stable truncated version of Waterman's T-matrix approach (for cost effective realization at each multiple particle configuration corresponding to the pseudospectral quadrature points) to simulate the statistical properties of the stochastic model. We demonstrate the efficiency of the algorithm for configurations with non-smooth and non-convex bodies with distinct material properties, and random locations and orientations with normal and log-normal distributions.

The parallel volume at large distances

In this paper we examine the asymptotic behavior of the parallel volume of planar non-convex bodies as the distance tends to infinity. We show that the difference between the parallel volume of the convex hull of a body and the parallel volume of the body itself tends to 0. This yields a new proof for the fact that a planar body can only have polynomial parallel volume if it is convex. Extensions to Minkowski spaces and random sets are also discussed.

Extension of Dirac's chord method to the case of a nonconvex set by the use of quasi-probability distributions

The Dirac's chord method may be suitable in different areas of physics for the representation of certain six-dimensional integrals for a convex body using the probability density of the chord length distribution. For a homogeneous model with a nonconvex body inside a medium with identical properties an analogue of the Dirac's chord method may be obtained, if to use so-called generalized chord distribution. The function is defined as normalized second derivative of the autocorrelation function. For nonconvex bodies this second derivative may have negative values and could not be directly related with a probability density. An interpretation of such a function using alternating sums of probability densities is considered. Such quasi-probability distributions may be used for Monte Carlo calculations of some integrals for a single body of arbitrary shape and for systems with two or more objects and such applications are also discussed in this work.

On the two-dimensional rotational body of maximal Newtonian resistance

We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that the bodies are moving forward and at the same time slowly rotating. A two-dimensional geometric shape that confers to the body a resistance very close to the theoretical supremum value is obtained, which improves previous results.

Exterior Monge–Ampère solutions

We discuss the Siciak–Zaharjuta extremal function of a real convex body in C n , a solution of the homogeneous complex Monge–Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R 2 .

Two-dimensional body of maximum mean resistance

A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, [Alexander Yu. Plakhov, Paulo D.F. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271–2287], shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement. Giving continuity to this study, new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier. This result, now presented, consists of a two-dimensional shape which confers on the body a resistance which is very close to its theoretical supremum value. But its interest does not lie solely in the maximization of Newtonian resistance; on regarding its characteristics, other areas of application are seen to begin to appear which are thought to be capable of having great utility. The optimal shape which has been encountered resulted from numerical studies, thus it is the object of additional study of an analytical nature, where it proves some important properties which explain in great part its effectiveness.