Approximate Implicitization using Linear Algebra

In this talk we consider a range of methods for exact and approximate implicitization of rational parametric curves and surfaces using linear algebra. The framework of numerical linear algebra provides a large family of methods for (approximate) implicitization which vary in terms of stability, approximation quality and speed of implementation. We compare various methods which work by minimizing the algebraic error and discuss the relative merits of each.Approximate Implicitization using Linear Algebr

Approximate Implicitization using Chebyshev Polynomials

Whereas traditional approaches to implicitization of rational parametric curves have focused on exact methods, the past two decades have seen increased interest in the application of approximate methods for implicitization. In this talk we will discuss how the properties of the Chebyshev polynomial basis can be used to improve the speed, stability and approximation quality of existing algorithms for approximate implicitization. We will also look at how the algorithm is well suited to parallelization

Approximate Implicitization and Approximate Null Spaces

Easy conversion of elementary curves and surfaces (lines, circles, ellipses, planes, spheres, cylinders, cones,. . . ) to rational parametric and implicit representations is central in many algorithms used in CAD-systems. For rational Bézier and NURBS-surfaces no such easy conversion exists, a rational parametric surface of bi-degree (n,k) has in the general case an algebraic degree of 2nk giving the bi-cubic Beziér surface a degree 18 implicit representation. Essential to approximate implicitization is the combination of a rational parametric surface p(s,t), (s,t) in&nbsp; [0,1]x[0,1].&nbsp;&nbsp;with the algebraic surface to be found q(x,y,z,h) = 0. The degree m of q, should satisfy 0 < m ≤ 2nk. The combination results in the following factorization, q(p(s,t))=a(s,t)Db, where b contains the unknown coefficients of q, and a(s,t) is an array that contains basis function represented in the tensor product Bernstein basis. Similar expressions exist for rational parametric curves and triangular Bézier surfaces. The smallest singular values of D&nbsp;and&nbsp;&nbsp;their respective coeffiicient vectors&nbsp; represent alternative implicit approximations to p(s,t). If m = 2nk we&nbsp;know that an exact solution exists and that the smallest singular value will be zero. The problem of finding an approximate algebraic representation of p(s,t) has been refomulated to a problem of finding an approximate null space of the matrix D. One obvious choice is Singular Value Decomposition, however, alternative direct elimination methods also exist.Approximate Implicitization and Approximate Null Space

Approximate Implicitization and Approximate Null Spaces

Easy conversion of elementary curves and surfaces (lines, circles, ellipses, planes, spheres, cylinders, cones,. . . ) to rational parametric and implicit representations is central in many algorithms used in CAD-systems. For rational Bézier and NURBS-surfaces no such easy conversion exists, a rational parametric surface of bi-degree (n,k) has in the general case an algebraic degree of 2nk giving the bi-cubic Beziér surface a degree 18 implicit representation. Essential to approximate implicitization is the combination of a rational parametric surface p(s,t), (s,t) in&nbsp; [0,1]x[0,1].&nbsp;&nbsp;with the algebraic surface to be found q(x,y,z,h) = 0. The degree m of q, should satisfy 0 < m ≤ 2nk. The combination results in the following factorization, q(p(s,t))=a(s,t)Db, where b contains the unknown coefficients of q, and a(s,t) is an array that contains basis function represented in the tensor product Bernstein basis. Similar expressions exist for rational parametric curves and triangular Bézier surfaces. The smallest singular values of D&nbsp;and&nbsp;&nbsp;their respective coeffiicient vectors&nbsp; represent alternative implicit approximations to p(s,t). If m = 2nk we&nbsp;know that an exact solution exists and that the smallest singular value will be zero. The problem of finding an approximate algebraic representation of p(s,t) has been refomulated to a problem of finding an approximate null space of the matrix D. One obvious choice is Singular Value Decomposition, however, alternative direct elimination methods also exist.Approximate Implicitization and Approximate Null Space

Trivariate Spline Representations for Computer Aided Design and Additive Manufacturing

Digital representations targeting design and simulation for Additive Manufacturing (AM) are addressed from the perspective of Computer Aided Geometric Design. We discuss the feasibility for multi-material AM for B-rep based CAD, STL, sculptured triangles as well as trimmed and block-structured trivariate locally refined spline representations. The trivariate spline representations support Isogeometric Analysis (IGA), and topology structures supporting these for CAD, IGA and AM are outlined. The ideas of (Truncated) Hierarchical B-splines, T-splines and LR B-splines are outlined and the approaches are compared. An example from the EC H2020 Factories of the Future Research and Innovation Actions CAxMan illustrates both trimmed and block-structured spline representations for IGA and AM

Reverse engineering of CAD models via clustering and approximate implicitization

In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive shapes by combining clustering analysis with approximate implicitization. The proposed method is automatic and can recover algebraic hypersurfaces of any degree in any dimension. In exact arithmetic, the algorithm returns exact results. All the required parameters, such as the implicit degree of the patches and the number of clusters of the model, are inferred using numerical approaches in order to obtain an algorithm that requires as little manual input as possible. The effectiveness, efficiency and robustness of the method are shown both in a theoretical analysis and in numerical examples implemented in Python

Parametric Shape Optimization for Combined Additive–Subtractive Manufacturing

In industrial practice, additive manufacturing (AM) processes are often followed by post-processing operations such as heat treatment, subtractive machining, milling, etc., to achieve the desired surface quality and dimensional accuracy. Hence, a given part must be 3D-printed with extra material to enable this finishing phase. This combined additive/subtractive technique can be optimized to reduce manufacturing costs by saving printing time and reducing material and energy usage. In this work, a numerical methodology based on parametric shape optimization is proposed for optimizing the thickness of the extra material, allowing for minimal machining operations while ensuring the finishing requirements. Moreover, the proposed approach is complemented by a novel algorithm for generating inner structures to reduce the part distortion and its weight. The computational effort induced by classical constrained optimization methods is alleviated by replacing both the objective and constraint functions by their sparse grid surrogates. Numerical results showcase the effectiveness of the proposed approach.acceptedVersio