Non-convex Polyhedra Research Articles

Optimization of Packing Irregular Three-Dimensional Objects

Introduction. Nowadays the irregular packing problem is becoming more important, since effective space management and optimal arrangement of objects are becoming key factors for ensuring efficiency and saving resources in a wide range of applications, e.g., additive manufacturing, space engineering, material sciences and logistics. It becomes an integral part of strategic development in the fields of production and science. The purpose of the paper. The paper is devoted to construction of a mathematical model and development of an efficient technique for densely filling a container with the maximum number of sets of irregular three-dimensional objects. Results. Irregular objects are approximated with a certain accuracy by non-convex polyhedra, which can be represented by the union of convex polytopes. Non-overlapping and containment constraints are described using quasi-phi-functions and phi-functions. A mathematical model of the packing problem is provided as a mixed-integer nonlinear programming considering given proportions of different types of objects. A solution strategy is proposed to search for local-optimal solutions. To find reasonable feasible packing, a fast algorithm based on a strip approximation of objects is used. A numerical example of the development of a print map of a set of industrial parts with maximum filling of the working chamber of a 3D-printer is given. Conclusions. The results confirm the efficiency of the proposed packing strategy, which is based on an integrated approach that takes into account the geometric features of irregular objects and their completeness. Keywords: packing, irregular objects, set of parts, mathematical modeling, optimization, 3D-printing.

Efficient Exact Enumeration of Single-Source Geodesics on a Non-Convex Polyhedron

In this paper, we consider enumeration of geodesics on a polyhedron, where a geodesic means locally-shortest path between two points. Particularly, we consider the following preprocessing problem: given a point [Formula: see text] on a polyhedral surface and a positive real number [Formula: see text], to build a data structure that enables, for any point [Formula: see text] on the surface, to enumerate all geodesics from [Formula: see text] to [Formula: see text] whose length is less than [Formula: see text]. First, we present a naive algorithm by removing the trimming process from the MMP algorithm (1987). Next, we present an improved algorithm which is practically more efficient on a non-convex polyhedron, in terms of preprocessing time and memory consumption. Moreover, we introduce a single-pair geodesic graph to succinctly encode a result of geodesic query. Lastly, we compare these naive and improved algorithms by some computer experiments.

De Rham compatible Deep Neural Network FEM

On general regular simplicial partitions T of bounded polytopal domains Ω⊂Rd, d∈{2,3}, we construct exact neural network (NN) emulations of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical “Raviart–Thomas element”, and the “Nédélec edge element”. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions T of Ω are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension d≥2. Our “FE-Nets” are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra Ω⊂R3. They are thus an essential ingredient in the application of e.g., the methodology of “physics-informed NNs” or “deep Ritz methods” to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the “Crouzeix–Raviart” elements and Hybridized, Higher Order (HHO) methods.

Radiothermal Emission of Nanoparticles with a Complex Shape as a Tool for the Quality Control of Pharmaceuticals Containing Biologically Active Nanoparticles.

It has recently been shown that the titer of the SARS-CoV-2 virus decreases in a cell culture when the cell suspension is irradiated with electromagnetic waves at a frequency of 95 GHz. We assumed that a frequency range in the gigahertz and sub-terahertz ranges was one of the key aspects in the "tuning" of flickering dipoles in the dispersion interaction process of the surfaces of supramolecular structures. To verify this assumption, the intrinsic thermal radio emission in the gigahertz range of the following nanoparticles was studied: virus-like particles (VLP) of SARS-CoV-2 and rotavirus A, monoclonal antibodies to various RBD epitopes of SARS-CoV-2, interferon-α, antibodies to interferon-γ, humic-fulvic acids, and silver proteinate. At 37 °C or when activated by light with λ = 412 nm, these particles all demonstrated an increased (by two orders of magnitude compared to the background) level of electromagnetic radiation in the microwave range. The thermal radio emission flux density specifically depended on the type of nanoparticles, their concentration, and the method of their activation. The thermal radio emission flux density was capable of reaching 20 μW/(m2 sr). The thermal radio emission significantly exceeded the background only for nanoparticles with a complex surface shape (nonconvex polyhedra), while the thermal radio emission from spherical nanoparticles (latex spheres, serum albumin, and micelles) did not differ from the background. The spectral range of the emission apparently exceeded the frequencies of the Ka band (above 30 GHz). It was assumed that the complex shape of the nanoparticles contributed to the formation of temporary dipoles which, at a distance of up to 100 nm and due to the formation of an ultrahigh strength field, led to the formation of plasma-like surface regions that acted as emitters in the millimeter range. Such a mechanism makes it possible to explain many phenomena of the biological activity of nanoparticles, including the antibacterial properties of surfaces.

An Automated Space-Based Graph Generation Framework for Building Energy Consumption Estimation

The 3D information in Building Information Modeling (BIM) has received significant interest for smart city applications. Recently, employing Industry Foundation Classes (IFC) for BIM in data-driven methods for Building Energy Consumption Estimation (BECE) has gained momentum because of the enriched geometric and semantic information. However, despite extensive studies on applying the IFC data in BECE analysis, employing the full potential of the BIM remains poor due to its complex data model and incompatibility with data-driven algorithms. This paper proposes a framework to extract accurate semantic, geometry, and topology information from the room-level (space) IFC schema by introducing new geo-computation algorithms to deal with these challenges. Additionally, we define a new topological weighted relationship between spaces in different stories by combining common geometry area with energy resistance value. Eventually, the proposed weighted space-based graph will be constructed to decrease the original complexity of the IFC model, and it is compatible with graph-based machine learning algorithms. The results are promising, with more than 90% accuracy in extracting the geometry information for the convex and non-convex polyhedron rooms and 100% accuracy in detecting vertical and horizontal adjacent rooms. This study confirms the proposed approach’s efficiency, accuracy, and feasibility for space-based BECE analysis.

On Ray Shooting for Triangles in 3-Space and Related Problems

We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in $R^3$, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in $R^3$, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. Our approach is based on the polynomial partitioning technique. For example, our ray-shooting algorithm processes a set of $n$ triangles in $R^3$ into a data structure for answering ray shooting queries amid the given triangles, which uses $O(n^{3/2+\varepsilon})$ storage and preprocessing, and answers a query in $O(n^{1/2+\varepsilon})$ time, for any $\varepsilon>0$. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly $n^{5/8}$. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer $m$ queries on $n$ objects in \[ \max \left\{ O(m^{2/3}n^{5/6+\varepsilon} + n^{1+\varepsilon}),\; O(m^{5/6+\varepsilon}n^{2/3} + m^{1+\varepsilon}) \right\} \] time, for any $\varepsilon>0$, again an improvement over the earlier bounds.

Аналоги теоремы Гаусса-Александрова для однородных ориентируемых невыпуклых многогранников с выпуклыми гранями

The paper defines the spherical image of a polyhedral angle of a uniform oriented nonconvex polyhedron, introduces the concept of curvature of realization of such an angle and proves the theorem about the equality of the area of the spherical image of a polyhedral angle with the feature ”fold” to its realization curvature.

Maximal regularity of multistep fully discrete finite element methods for parabolic equations

AbstractThis article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

Multistage Approach to Solving the Optimization Problem of Packing Nonconvex Polyhedra

The paper considers the problem of packing nonconvex polyhedra into a container of minimum volume. An exact mathematical model of the problem of packing nonconvex polyhedra with continuous translations and rotations is constructed. Characteristics of the mathematical model are analyzed and are used as the basis to develop a multistage solution approach to obtain a nearly optimal solution, which is not the global minimum but is a proved local minimum. Numerical examples are given.

Secondary Polytope and Secondary Power Diagram

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.

Hopper flow of irregularly shaped particles (non-convex polyhedra): GPU-based DEM simulation and experimental validation

Numerous practical applications of the Discrete Element Method (DEM) require a flexible description of particles that can account for irregular and non-convex particle shape features. Capturing the particle non-convexity is important since it allows to model the physical interlocking when the particles are in contact. To that end, the most flexible approach to capture the particle shape is via a polyhedron, which provides a faceted representation of any shape, albeit at a significant computational cost. In this study we present a decomposition approach to modeling non-convex polyhedral particles as an extension of an existing open source convex polyhedral discrete element code, BlazeDEM-GPU, which computes using general purpose graphical processing units (GPGPUs). Although the principle of decomposition of non-convex particles into convex particles is not new, its application by the discrete element modeling community has been rather limited. The non-convex extension of BlazeDEM-GPU was validated using a hopper flow experiment with identical convex and identical non-convex 3D printed particles. The experiment was designed around two sensitive flow points, with the convex particles following the intermittent flow and the non-convex particles forming stable arches. It was demonstrated that the DEM simulations can be applied to reproduce both the convex and the non-convex flow behavior using the same parameter set. This study is a significant step towards general computing of non-convex particles for industrial-scale applications using the GPGPUs.

Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra

In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the $L^q$ norm, $1\leq q\leq\infty$, and the maximal $L^p$-regularity of semi-discrete finite element solutions of parabolic equations are proved. By using these results, the problem of maximum-norm stability of the finite element parabolic projection is reduced to the maximum-norm stability of the Ritz projection, which currently is known to hold for general polygonal domains and convex polyhedral domains.

On tetrahedralisations of generalised Chazelle polyhedra with interior Steiner points

In 1984, Chazelle constructed a family of non-convex polyhedra which can be parametrised by the number N of the defining line segments and the vertical shift ε between the two saddle surfaces. Such a polyhedron is known as the Chazelle polyhedron (Chazelle, 1984). It establishes a quadratic lower bound on the minimum number of convex pieces for the 3d convex decomposition problem. Chazelle polyhedra are not decomposable without using addition points, so-called Steiner points. In this paper, we study the problem of triangulating a Chazelle polyhedron with only interior Steiner points. It is motivated by a crucial step in tetrahedral mesh generation in which a set of arbitrary constraints (edges or faces) need to be entirely preserved. We first “cut off” the volume of the Chazelle polyhedron by removing the regions that can be tetrahedralised without adding Steiner points. This leads to a 3d non-convex polyhedron whose vertices belong to the two shifted saddle surfaces. We call it the reduced Chazelle polyhedron. We then give a set of (N+1)2 interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron with 4(N+1) input line segments. The proof uses an intuitive relation between a 3d tetrahedralisation and a sequence of 2d flips. In particular, we design an edge splitting and flipping algorithm and prove that it will result a tetrahedralisation of the reduced Chazelle polyhedron. Finally, we construct a family of polyhedra, so-called generalised Chazelle polyhedra in which the set of reduced Chazelle polyhedra is a subset of it. We prove that our placement of interior Steiner points also applies to tetrahedralise generalised Chazelle polyhedra.

Conservative and bounded volume-of-fluid advection on unstructured grids

This paper presents a novel Eulerian–Lagrangian piecewise-linear interface calculation (PLIC) volume-of-fluid (VOF) advection method, which is three-dimensional, unsplit, and discretely conservative and bounded. The approach is developed with reference to a collocated node-based finite-volume two-phase flow solver that utilizes the median-dual mesh constructed from non-convex polyhedra. The proposed advection algorithm satisfies conservation and boundedness of the liquid volume fraction irrespective of the underlying flux polyhedron geometry, which differs from contemporary unsplit VOF schemes that prescribe topologically complicated flux polyhedron geometries in efforts to satisfy conservation. Instead of prescribing complicated flux-polyhedron geometries, which are prone to topological failures, our VOF advection scheme, the non-intersecting flux polyhedron advection (NIFPA) method, builds the flux polyhedron iteratively such that its intersection with neighboring flux polyhedra, and any other unavailable volume, is empty and its total volume matches the calculated flux volume. During each iteration, a candidate nominal flux polyhedron is extruded using an iteration dependent scalar. The candidate is subsequently intersected with the volume guaranteed available to it at the time of the flux calculation to generate the candidate flux polyhedron. The difference in the volume of the candidate flux polyhedron and the actual flux volume is used to calculate extrusion during the next iteration. The choice in nominal flux polyhedron impacts the cost and accuracy of the scheme; however, it does not impact the methods underlying conservation and boundedness. As such, various robust nominal flux polyhedron are proposed and tested using canonical periodic kinematic test cases: Zalesak's disk and two- and three-dimensional deformation. The tests are conducted on the median duals of a quadrilateral and triangular primal mesh, in two-dimensions, and on the median duals of a hexahedral, wedge and tetrahedral primal mesh, in three-dimensions. Comparisons are made with the adaptation of a conventional unsplit VOF advection scheme to our collocated node-based flow solver. Depending on the choice in the nominal flux polyhedron, the NIFPA scheme presented accuracies ranging from zeroth to second order and calculation times that differed by orders of magnitude. For the nominal flux polyhedra which demonstrate second-order accuracy on all tests and meshes, the NIFPA method's cost was comparable to the traditional topologically complex second-order accurate VOF advection scheme.

An algorithm for determining the optimal variant of a cut gem with maximal mass and specified symmetry deviations

Denis S. Kokorev - Doctoral Student, Laboratory of Distributed Computational Systems, Institute for Information Transmission Problems RAS Address: 19, build.1, Bolshoy Karetny Per., Moscow, 127051, Russian FederationE-mail: korvin-d@yandex.ru The article discusses the problem of finding a polyhedron given shape inside another nonconvex polyhedron. This problem is a particular case of the 18th Hilbert problem, third part. It has a practical application in computer simulation of three-dimensional objects, moving autonomous robots, and the jewelry industry. The author uses this mathematical problem to find the facets of gemstones in uncut stones. The article offers a method for finding inscribed polyhedrons based on the reduction of the problem to a nonlinear programming problem and its solutions using ready-made software. The basic idea is that it is easy to describe this problem in terms of non-linear programming. Internal polyhedron volume is an objective function. Restrictions include the preservation of the combinatorial structure, one polyhedron standing inside another one, convexity, plus additional constraints necessary for practical purposes. The article describes two implementations of the algorithm: a client-server application and a local application. Their advantages and disadvantages are discussed. The algorithm is described not only in a mathematical point of view; some of its practical characteristics are also demonstrated. Compared to the previous article, the author has added a method that allows for solving the nonconvex case of a problem. This is a significant step forward from a mathematical point of view. In addition, it allows us to use the algorithm at all stages of gem cutting. The end of the article describes current evaluations of the effectiveness and running time, including on weak processors, and it offers plans for further development of the algorithm.This research was supported by the Russian Scientific Foundation (project No. 16-11-10352

An algorithm for determining the optimal variant of a cut gem with maximal mass and specified symmetry deviations

Denis S. Kokorev - Doctoral Student, Laboratory of Distributed Computational Systems, Institute for Information Transmission Problems RAS Address: 19, build.1, Bolshoy Karetny Per., Moscow, 127051, Russian FederationE-mail: korvin-d@yandex.ru The article discusses the problem of finding a polyhedron given shape inside another nonconvex polyhedron. This problem is a particular case of the 18th Hilbert problem, third part. It has a practical application in computer simulation of three-dimensional objects, moving autonomous robots, and the jewelry industry. The author uses this mathematical problem to find the facets of gemstones in uncut stones. The article offers a method for finding inscribed polyhedrons based on the reduction of the problem to a nonlinear programming problem and its solutions using ready-made software. The basic idea is that it is easy to describe this problem in terms of non-linear programming. Internal polyhedron volume is an objective function. Restrictions include the preservation of the combinatorial structure, one polyhedron standing inside another one, convexity, plus additional constraints necessary for practical purposes. The article describes two implementations of the algorithm: a client-server application and a local application. Their advantages and disadvantages are discussed. The algorithm is described not only in a mathematical point of view; some of its practical characteristics are also demonstrated. Compared to the previous article, the author has added a method that allows for solving the nonconvex case of a problem. This is a significant step forward from a mathematical point of view. In addition, it allows us to use the algorithm at all stages of gem cutting. The end of the article describes current evaluations of the effectiveness and running time, including on weak processors, and it offers plans for further development of the algorithm.This research was supported by the Russian Scientific Foundation (project No. 16-11-10352

Automatic Synthesis of Switching Controllers for Linear Hybrid Systems

We consider the problem of computing the controllable region of a Linear Hybrid Automaton with controllable and uncontrollable transitions, w.r.t. a reachability objective. We provide an algorithm for the finite-horizon version of the problem, based on computing the set of states that must reach a given non-convex polyhedron while avoiding another one, subject to a polyhedral constraint on the slope of the trajectory. Experimental results are presented, based on an implementation of the proposed algorithm on top of the tool SpaceEx.

Interpolation error estimates for harmonic coordinates on polytopes

Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.

On Tetrahedralisations of Reduced Chazelle Polyhedra with Interior Steiner Points

The non-convex polyhedron constructed by Chazelle, known as the Chazelle polyhedron[4], establishes a quadratic lower bound on the minimum number of convex pieces for the 3d polyhedron partitioning problem. In this paper, we study the problem of tetrahedralising the Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in tetrahedral mesh generation in which a set of arbitrary constraints (edges or faces) need to be entirely preserved. The goal of this study is to gain more knowledge about the family of 3d indecomposable polyhedra which needs additional points, so-called Steiner points, to be tetrahedralised. The requirement of only using interior Steiner points for the Chazelle polyhedron is extremely challenging. We first “cut off” the volume of the Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d non-convex polyhedron whose vertices are all in the two slightly shifted saddle surfaces which are used to construct the Chazelle polyhedron. We call it the reduced Chazelle polyhedron. It is an indecomposable polyhedron. We then give a set of (N + 1)2 interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron with 4(N + 1) vertices. The proof is done by transforming a 3d tetrahedralisation problem into a 2d edge flip problem. In particular, we design an edge splitting and flipping algorithm and prove that it gives to a tetrahedralisation of the reduced Chazelle polyhedron.

Analysis of a continuous finite element method for $$H(\mathrm{curl},\mathrm{div})$$ -elliptic interface problem

In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local $$L^2$$ projector to the curl operator and a pseudo-local $$L^2$$ projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate $$\mathcal{O }(h^r)$$ in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in $$\prod _{l=1}^L (H^r(\Omega _l))^3$$ for some $$r\in (1/2,1]$$ due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in $$\Omega =\bigcup _{l=1}^L \Omega _l$$ .