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A buckling analysis of functionally graded plates of a complex form resting on an elastic foundation and subjected to an in-plane nonuniform loading is performed by the R-functions method for the first time. The mathematical formulation of the problem is presented within the framework of the classical laminate plate theory. The plates considered consist of three layers. The middle layer (core) is ceramic or metal, a face layers are fabricated of functionally graded materials (FGMs). The power-law distribution of volume fraction of constituents is used to compute the effective material properties of FGM layers. The approach proposed and the software developed consider the heterogeneous subcritical state of the plates. First, the problem of in-plane elasticity problem is solved, and then the stability problem is considered. To solve both the problems, the Ritz method combined with the R-functions theory is used. The method proposed and the software developed are verified by comparing the buckling loads of square plates subjected to a nonuniform loading. The critical loads for sandwich FG plates of a complex geometry in a nonuniform edge compression are calculated. The effects of boundary conditions, the scheme of layer arrangement, and the type of FGM on the critical load are studied. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for analyzing the free vibration response of variable-stiffness panels. A method is proposed herein, which relies upon a combination of the R-functions and the Ritz method. A powerful and peculiar aspect of this approach consists in the possibility of handling any complex geometry with reduced modeling effort and few degrees of freedom to be handled. The so-obtained formulation is of particular interest for the early design studies of plate- and shell-like structures, especially when topological features are of concern. Specifically, the possibility of studying configurations of arbitrary shape with reduced effort is of crucial importance for exploiting the potential offered by composites with nonstraight fibers. Exemplary results are presented for structures with different geometries and shapes. The comparison against results from the literature and finite element calculations reveals the potential of the approach as a valuable mean for assisting the design of innovative variable-stiffness configurations. [ABSTRACT FROM AUTHOR]

In this paper, based on the R-functions theory, methods have been developed and equations have been constructed for the 3D printing of hex-head screws with Bristol, Pentalobe, Polydrive and other types of screw slots. Such screws are used both in personal computers and other high-end equipment. The Bristol slot has four or six radial grooved beams. The advantage of the design of this slot is the correct perpendicular, rather than tangential, vector of force application when the slot is rotated by a tool, which minimizes the risk of stripping out the slot. For this reason, the Bristol slot is used in soft metal screws. Compared to the internal hex, the Bristol slot allows a noticeably higher torque, only slightly higher than that of the Torx slot. This type of slot is used in aviation, high-end telecommunications equipment, cameras, air brakes, agricultural equipment, astronomical equipment, and foreign military equipment. Variations with a pin in the center are found in game consoles to prevent the use of a flat-blade screwdriver as an improvised key. The Pentalobe slot is a five-point slot designed by Apple and used in its products to limit unauthorized disassembly. It was first used in mid 2009 to mount MacBook Pro batteries. Its miniature version was used in the iPhone 4 and later models, in the MacBook Air (available since late 2010 models), and the MacBook Pro with Retina screens. The Polydrive slot is a starlike slot with rounded star points, used in the automotive industry for applications requiring high tightening torque. The Torq-set slot is a cross slot for fasteners requiring high tightening torque. The grooves are slightly offset, not intersecting at one point. Fasteners with this type of slot are used in military aviation, for example, in E-3, P-3, F-16, Airbus, Embraer, and Bombardier Inc. The Phillips Screw Company owns the trademark and manufactures fasteners with this type of slot. The slot design standards are National Aerospace Standard NASM 3781 and NASM 4191 for the ribbed version. The resulting equations for the surfaces of screws were checked during the modeling of the screws before 3D printing. The 3D printing technology allows us to reduce the cost and labor intensity of manufacturing products, including complex slot screws. The analytical recording of designed objects makes it possible to use alphabetic geometric parameters, complex superposition of functions, which, in turn, allows us to quickly change their design elements. The positivity property of the constructed functions at the internal points of an object is very convenient for the implementation of 3D printing

Creation of mathematical models of objects to be 3D printed is of considerable interest, which is associated with the active introduction of 3D printing in various industries. The advantages of using modern 3D printers are: lower production costs and shorter periods of time for their appearance on the market, modeling objects of any shape and complexity, speed and high precision of manufacturing, their ability to use various materials. One of the methods for solving the problem of creating a mathematical and computer model of the object being designed is the application of the R-functions theory, with the help of which it is possible to describe geometric objects of complex shapes in a single analytical expression. The use of alphabetic parameters, when one specifies geometric information in analytical form, allows one to quickly change the size and shape of the object being designed, which helps to spend less time on building computational models. The proposed method can significantly reduce the complexity of work in CAD systems in those cases when one needs to view a large number of design options in search of an optimal solution. This gives a great effect on reducing labor intensity in the construction of computational models to determine aero-gas-dynamic and strength characteristics. Characterization is also often associated with the need to account for changes in aircraft shape. This leads to the fact that the determination of aerodynamic characteristics only due to the need to build a large number of computational models increases the duration of work by months. With parametric assignments, computational regions change almost instantly. In this paper, on the basis of the basic apparatus of the theory of R-functions as well as cylindrical, spherical, ellipsoidal, and conusoidal support functions, a multiparametric equation for the surface of a Soyuz-Apollo spacecraft model is constructed. A number of support functions were normalized according to a general formula, which made it possible to illustrate a new approach to constructing three-dimensional equations for surfaces of a given thickness.

Unmanned aerial vehicles (UAVs) are becoming increasingly demanded worldwide. The scope of their use is very extensive. They are used for military purposes, delivery of goods, environmental monitoring, border patrolling, aerial reconnaissance and mapping, traffic control, etc. UAVs have a number of important advantages over manned aircraft. These advantages include relatively low costs of UAVs at their long flight durations and ranges, their low operating costs, and the ability to perform maneuvers with overloads that exceed the physical capabilities of a human being, making their development more active. One cannot imagine the designing of UAVs and control systems without mathematical modeling. To build mathematical models, high-speed computers and modern software tools have been created, Solid Works, Ansys CFX, POLYE software systems being among them. There arises a problem of specifying and quickly changing geometric information to create a mathematical and computer model of the UAV being designed. At the design stage, there can be solved a lot of tasks that are put before researchers as regards the use of UAVs. At the same time, insufficient attention is paid to the parametric representation of aircraft surfaces. Expanding the scope of using the apparatus of the theory of R-functions for modeling UAV surfaces is an urgent scientific and technical task. In this paper, for the first time, using the theory of R-functions, we build up the equation of the surface of a flying wing UAV in the form of a single analytical expression with alphabetic parameters. This equation can be used in solving various practical problems as well as developing and manufacturing the product itself, for example, on a 3D printer. The proposed method for specifying the shapes of products by using a limited number of parameters can significantly reduce the complexity of work in CAD systems in cases where it is required to view a large number of design options in search of an optimal solution. In this paper, we build a 14-parameter family of flying wing UAV surfaces. By changing the values of alphabetic parameters, we can quickly explore its various forms.

The free vibration of plates and shallow shells with/without cutouts made of functionally graded materials (FGM) is investigated using variational FG shallow shells with temperature dependent mechanical characteristics of the constituent materials. First‐order shear deformation theory of shallow shells is employed. It is supposed that material properties vary through thickness according to a power‐law distribution of the constituent's volume fraction. They depend on both the temperature and the thickness. Temperature field is modeled by one‐dimensional heat transfer equation, since the temperature is varied only in thickness direction. Solution of this equation is determined by a polynomial power series expansion. Corresponding software was developed to implement the proposed approach. The vibration analysis was carried out for FG plates and shallow shells with cutout and various boundary conditions. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of boundary equations for geometric domains with perforation. Different types of perforated geometric domains are considered. The R-functions method for analytical modelling of perforated geometrical domains is used. For all constructed equations, function plots are obtained.

This paper deals with the construction of boundary equations for geo- metric domains with perforation. Different types of perforated geometric domains are considered. The R-functions method for analytical modelling of perforated geo- metrical domains is used. For all constructed equations, function plots are obtained. [ABSTRACT FROM AUTHOR]

Based on the R-functions theory, new approaches to analytical identification of drone surfaces for realization of 3D printing technology have been developed. The R-functions theory allows one to describe geometric objects of a complex shape with a single analytical expression, that is, obtain a mathematical model of the object in a form of an equation. To derive such equations, we used both the well-known standard primitive (sphere, ellipsoid, cylinder, cone, pyramid, etc.) procedure and a new approach, blending on a frame, which enables derivation of multiparameter equations with specified properties. Multiparameter equations of surfaces of drones of various types and purposes have been derived and visualized. Adequacy of the results to the designed objects was confirmed by visualization, both in conditions of operation of the RFPreview program and by realization on a 3D printer. The use of literal parameters when specifying geometric information in an analytical form makes it possible to promptly change size and shape of the designed objects which helps reduce time required to build computational models. The proposed method can reduce labor input in operation of CAD systems by months in cases when it is necessary to view a large number of design variants in a search for an optimal solution. Having the object equation, one can easily obtain equation of any of its sections which is useful for numerical calculations, namely, when building computational meshes. This can have a great effect on reducing complexity in construction of computational models for determining aero-gas-dynamic and strength characteristics. Characterization is also often associated with the need to account for changes in the aircraft shape. This leads to the fact that establishment of aerodynamic characteristics just because of the need to build a large number of computational models to account for this factor increases work duration by months. When specifying parameters, change of the rated operating conditions is made almost instantly. [ABSTRACT FROM AUTHOR]

The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO. [ABSTRACT FROM AUTHOR]

In this paper, the process of convection in the Earth’s mantle in the presence of a floating continent is considered. The model is a two-dimensional rectangular region of viscous thermally conducting fluid that obeys the equations of hydrodynamics. The method of Rvachev R-functions is used for the description of the problem geometry and the boundary conditions. [ABSTRACT FROM AUTHOR]

The mathematical content of the algorithm for calculating motion data along the plane of the mobile robot taking into account its bounding circle and the obstacles in the form of figures belonging to this plane is offered. The mathematical model is based on the use of elements of the theory of R-functions and virtual potential functions.

The article deals with a flow in the tube of rectangular cross-section with an inner circular cylindrical element. We use the numerical method basing on R-functions. This method is meshfree and therefore more efficient than the finite element method that requires remeshing when the geometry of problem is changed. The dependence of the flow on the diameter of the central cylinder and its position in the duct is investigated under constant pressure gradient. It was found that the resistance decreases if the inner element is moved from the center of the duct. [ABSTRACT FROM AUTHOR]

Mesh-free methods allow one to perform engineering calculations directly via a geometrical model constructed in the CAD system. They help to avoid the difficulties and errors in generation of a finite element mesh. In the present work, the most suitable techniques for analysis of complex solid-state constructions of polymeric composites are selected among mesh-free methods. A computation algorithm is also developed on the basis of these methods. [ABSTRACT FROM AUTHOR]

In this paper, we devise cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes. In the variational scheme to obtain max-ent basis functions, the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in R 2 . More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighbor-ring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised. The properties and mathematical foundations of the new compactly-supported approximants are described, and its use to solve two-dimensional elliptic boundary-value problems (Poisson equation and linear elasticity) is demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms are established. [ABSTRACT FROM AUTHOR]

In problems with evolving boundaries, interfaces or cracks, blending functions are used to enrich the underlying domain with the known behavior on the enriching entity. The blending functions are typically dependent on the distance from the propagating boundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This may require either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary. In this paper a purely algebraic, and computationally efficient technique is described for constructing distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non-robust distance calculation. The constructed distance measures are level sets of the implicitized constituent Bezier patches of the NURBS surfaces that are obtained purely algebraically. Since, in general, the implicitized functions extend beyond the parametric range of the generating Bezier patch, algorithmic procedures are developed to trim these global implicit functions to the boundaries of the Bezier patch. Boolean compositions are then carried out between adjoining Bezier patches to construct a composite distance field over the domain. The compositions rely on R-functions that are also algebraic in nature. The developed technique is demonstrated by constructing algebraic distance field for complex geometries and by solving a variety of examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient and flexible design method that integrates the B-spline finite cell method (B-spline FCM) and the level set function (LSF) for stress constrained shape and topology optimization. Any structure of complex geometry is embedded within an extended, regular and fixed Eulerian mesh no matter how the structure is optimized. High-order B-spline shape functions are further implemented to ensure precisions of stress analysis and sensitivity analysis. Meanwhile, level set functions, i.e., implicit functions are used to enable topological changes of the considered structure through smooth boundary variations. Involved parameters rather than the conventional discrete form of LSF are directly taken as design variables to facilitate the numerical computing process. To be specific, the LSF is constructed by means of R-functions that incorporate cubic splines as implicit functions to offer flexibilities for shape optimization within the framework of fixed mesh, while the compactly supported radial basis functions (CS-RBFs) are employed as implicit functions for stress constrained topology optimization. It is shown the proposed FCM/LSF method is a convenient approach that makes it possible to calculate stress and stress sensitivities with high precision. Representative examples of shape and topology optimization with and without stress constraints are solved with success demonstrating the advantages of the FCM/LSF method. [ABSTRACT FROM AUTHOR]

In this paper we completely describe functions generating the infinite totally nonnegative Hurwitz matrices. In particular, we generalize the well-known result by Asner and Kemperman on the total nonnegativity of the Hurwitz matrices of real stable polynomials. An alternative criterion for entire functions to generate a Pólya frequency sequence is also obtained. The results are based on a connection between a factorization of totally nonnegative matrices of the Hurwitz type and the expansion of Stieltjes meromorphic functions into Stieltjes continued fractions (regular $$C$$ -fractions with positive coefficients). [ABSTRACT FROM AUTHOR]

Abstract: Gielis curves (GC) can represent a wide range of shapes and patterns ranging from star shapes to symmetric and asymmetric polygons, and even self intersecting curves. Such patterns appear in natural objects or phenomena, such as flowers, crystals, pollen structures, animals, or even wave propagation. Gielis curves and surfaces are an extension of Lamé curves and surfaces (superquadrics) which have benefited in the last two decades of extensive researches to retrieve their parameters from various data types, such as range images, 2D and 3D point clouds, etc. Unfortunately, the most efficient techniques for superquadrics recovery, based on deterministic methods, cannot directly be adapted to Gielis curves. Indeed, the different nature of their parameters forbids the use of a unified gradient descent approach, which requires initial pre-processings, such as the symmetry detection, and a reliable pose and scale estimation. Furthermore, even the most recent algorithms in the literature remain extremely sensitive to initialization and often fall into local minima in the presence of large missing data. We present a simple evolutionary algorithm which overcomes most of these issues and unifies all of the required operations into a single though efficient approach. The key ideas in this paper are the replacement of the potential fields used for the cost function (closed form) by the shortest Euclidean distance (SED, iterative approach), the construction of cost functions which minimize the shortest distance as well as the curve length using R-functions, and slight modifications of the evolutionary operators. We show that the proposed cost function based on SED and R-function offers the best compromise in terms of accuracy, robustness to noise, and missing data. [Copyright &y& Elsevier]

Abstract: The AGENT simulation system is used for detailed three-dimensional modeling of neutron transport and corresponding properties of nuclear reactors of any design. Numerical solution to the neutron transport equation in the AGENT system is based on the Method of Characteristics (MOCs) and the theory of R-functions. The latter of which is used for accurately describing current and future heterogeneous lattices of reactor core configurations. The AGENT code has been extensively verified to assure a high degree of accuracy for predicting neutron three-dimensional point-wise flux spatial distributions, power peaking factors, reaction rates, and eigenvalues. In this paper, a new AGENT code feature, a computational steering, is presented. This new feature provides a novel way for using deterministic codes for fast evaluation of reactor core parameters, at no loss to accuracy. The computational steering framework as developed at the Technische Universität München is smoothly integrated into the AGENT solver. This framework allows for an arbitrary interruption of AGENT simulation, allowing the solver to restart with updated parameters. One possible use of this is to accelerate the convergence of the final values resulting in significantly reduced simulation times. Using this computational steering in the AGENT system, coarse MOC resolution parameters can initially be selected and later update them – while the simulation is actively running – into fine resolution parameters. The utility of the steering framework is demonstrated using the geometry of a research reactor at the University of Utah: this new approach provides a savings in CPU time on the order of 50%. [Copyright &y& Elsevier]

Abstract: Integrated design of the structure topology and involved component layout is a challenging design issue when compared with traditional topology optimization. In this paper, we propose an implicit modeling approach that works completely on an Eulerian finite element mesh throughout the whole optimization process. To this aim, implicit level-set functions and R-functions are employed to describe geometrical shapes of movable components. In particular, a modified arctan function is adopted to depict the material discontinuity along the interface between the structure domain and each component domain. They are then used for material interpolations and analytical sensitivity analysis w.r.t. both pseudo-density design variables and location design variables related to the host structure and components, respectively. Based on a variety of numerical tests, it is demonstrated that considered design problems with movable components can easily be solved by extending the SIMP material model based topology optimization approach using an Eulerian mesh and the gradient-based optimization algorithm. [Copyright &y& Elsevier]

A method for determining the optimal internal parameters of technical systems, based on the analytical description of the operational capability range, which is defined as a set of linear constraints, is considered. An expression for the objective function, which allows one to use any known search method for optimizing the system by the criterion of the operational capability reserve, is derived. [ABSTRACT FROM AUTHOR]

We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde, and other structured matrices, Stietjes- and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results. [ABSTRACT FROM AUTHOR]

Abstract: The popularity of medial axis in shape modeling and analysis comes from several of its well known fundamental properties. For example, medial axis captures the connectivity of the domain, has a lower dimension than the space itself, and is closely related to the distance function constructed over the same domain. We propose the new concept of a medial zone of an n-dimensional semi-analytic domain that subsumes the medial axis of the same domain as a special case, and can be thought of as a ‘thick’ skeleton having the same dimension as that of . We show that by transforming the exact, non-differentiable, distance function of domain into approximate but differentiable distance functions, and by controlling the geodesic distance to the crests of the approximate distance functions of domain , one obtains families of medial zones of that are homeomorphic to the domain and are supersets of . We present a set of natural properties for the medial zones of and discuss practical approaches to compute both medial axes and medial zones for 3-dimensional semi-analytic sets with rigid or evolving boundaries. Due to the fact that the medial zones fuse some of the critical geometric and topological properties of both the domain itself and of its medial axis, re-formulating problems in terms of medial zones affords the ‘best of both worlds’ in applications such as geometric reasoning, robotic and autonomous navigation, and design automation. [Copyright &y& Elsevier]

This article introduces the use of R-functions to compose single Lyapunov functions (LFs) via classic Boolean operators, with the aim to obtain a rich family of non-conventional, generally non-convex functions. The main benefit of the proposed composition is the nice geometric interpretation, since it corresponds to intersection and union operations in the phase space region. The composition of LFs is parameterised through a variable γ and classic compositions of LFs through min and max operations are recovered as a special case for a particular value of γ. The proposed logical composition is applied to region of asymptotic stability (RAS) estimation problems, where the union of several LFs corresponds to the union of the RAS estimates obtained from the separate use of each LF. Likewise, the intersection of several LFs defined on independent subsets of the state space variables provides a single LF for the overall dynamical system. Sufficient conditions for the composition function to be an LF are provided and results are described through several examples of classic nonlinear dynamical systems. [ABSTRACT FROM AUTHOR]

general method is proposed to derive equations of irregular curves O ( x, y) = 0 and of irregular surfaces O ( x, y, z) =0 in implicit form, where the functions O ( x, y) and O ( x, y, z) belong to a prescribed differentiability class. The method essentially involves interlineation and interflation of functions. An example is considered. [ABSTRACT FROM AUTHOR]

The paper proposes a method to solve geometrically nonlinear bending problems for thin orthotropic shallow shells and plates interacting with a Winkler-Pasternak foundation under transverse loading. This method is based on Ritz's variational method and the R-function method. The developed algorithm and software are used to solve a number of test problems and to study complex-shaped shells. The effect of the shape of shells, the boundary conditions, the stiffness of the foundation, and the load distribution on the behavior of isotropic and orthotropic shells undergoing geometrically nonlinear bending is studied [ABSTRACT FROM AUTHOR]

Abstract: Shape skeletons are fundamental concepts for describing the shape of geometric objects, and have found a variety of applications in a number of areas where geometry plays an important role. Two types of skeletons commonly used in geometric computations are the straight skeleton of a (linear) polygon, and the medial axis of a bounded set of points in the -dimensional Euclidean space. However, exact computation of these skeletons of even fairly simple planar shapes remains an open problem. In this paper we propose a novel approach to construct exact or approximate (continuous) distance functions and the associated skeletal representations (a skeleton and the corresponding radius function) for solid 2D semi-analytic sets that can be either rigid or undergoing topological deformations. Our approach relies on computing constructive representations of shapes with R-functions that operate on real-valued halfspaces as logic operations. We use our approximate distance functions to define a new type of skeleton, i.e, the C-skeleton, which is piecewise linear for polygonal domains, generalizes naturally to planar and spatial domains with curved boundaries, and has attractive properties. We also show that the exact distance functions allow us to compute the medial axis of any closed, bounded and regular planar domain. Importantly, our approach can generate the medial axis, the straight skeleton, and the C-skeleton of possibly deformable shapes within the same formulation, extends naturally to 3D, and can be used in a variety of applications such as skeleton-based shape editing and adaptive motion planning. [Copyright &y& Elsevier]

Normalized equations for regular no-gons translated nd times in a regular nb-gon are set up based on coordinate transformations that allow translating geometric objects on segments. Such approaches are especially important in solving automation problems in solid modeling, geometric design, and boundary-value problems of mathematical physics. [ABSTRACT FROM AUTHOR]

Abstract: Most solid models are archived using boundary representations, but they are created, edited, and optimized using high level constructive methods that rely on parameterized Boolean set operations and feature-based techniques. Downstream applications often require optimization of integral-valued performance measures over such models that include volume, mass, and energy properties, as well as more general distributed fields (stress, temperature, etc.). A key computational utility in all such applications is the computation of the sensitivity of the performance measure with respect to the parameters in the solid''s construction history. We show that for a class of performance measures defined as domain integrals, the sensitivity with respect to a parameter requires integration over a subset of the solid''s boundaries that is affected by that parameter. In contrast to earlier methods, the proposed approach for computing sensitivities does not require solid''s boundary to remain homeomorphic, and may be used with most types of constructive representations, including CSG and feature-based representations, where the defining Boolean expression may not be known. The simplicity and effectiveness of the proposed technique are illustrated on several common shape optimization problems. [Copyright &y& Elsevier]

We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a Constructive Solid Geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives' parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives' implicit equations using R-functions. Depending on the chosen R-function, this equation is continuous and can be differentiable everywhere. The primitives' parametric representations are used to directly polygonize the resulting surface by generating vertices that belong exactly to the zero-set of the resulting implicit equation. The proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression. Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling. [ABSTRACT FROM AUTHOR]

. In a meshfree system, a geometric model of a domain neither conforms to, nor is restricted by a spatial discretization. Such systems for engineering analysis offer numerous advantages over the systems that are based on traditional mesh-based methods, but they also require radical approaches to enforcing boundary conditions and novel computational tools for differentiation, integration, and visualization of fields and solutions. We show that all of these challenges can be overcome, and describe SAGE (Semi-Analytic Geometry Engine) – a successful system specifically intended for meshfree engineering analysis. Our approach and individual modules are based on Rvachev’s Function Method (RFM) but the described techniques, algorithms, and software are applicable to all mesh-based and meshfree methods and have broad use beyond solutions of boundary value problems. [ABSTRACT FROM AUTHOR]