Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improv... more Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the top-of-the-line one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, convolution with PSIAC filters reduces to a stable inner product with the DG output and hence provides a stable and efficient implementation. The paper focuses on the remarkable fact that, for the canonical linear or non-linear hyperbolic test equation, new piecewise constant PSIAC filters of small support outperform the top-of-the-line boundary filters in the literature. These least-degree filters reduce the error at the boundaries to less than the error even of the symmetric filters of the same support. Due to their simplicity, and since this least degree filter has an exact symbolic form, convolution is stable as well as efficient and derivatives of the convolved output are easy to compute.
Convolving the output of Discontinuous Galerkin computations with symmetric Smoothness-Increasing... more Convolving the output of Discontinuous Galerkin computations with symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. This paper interprets these filters as instances of a general class of position-dependent (PSIAC) spline filters that can have non-uniform knot sequences and skip B-splines of the sequence. PSIAC filters with rational knot sequences have rational coefficients. For prototype knot sequences , such as integer sequences that may have repeated entries, PSIAC filters can be expressed in symbolic form. Based on the insight that filters for shifted or scaled knot sequences are easily derived by non-uniform scaling of one prototype filter, a single filter can be re-used in different locations and at different scales. Computing a value of the convolution then simplifies to forming a scalar product of a short vector with the local output data. Restating one-sided filters in this form improves both stability and efficiency compared to their original formulation via numerical integration. PSIAC filtering is demonstrated for several established and one new boundary filter.
We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected e... more We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected edge graph consisting of only convex edges into a collection of topological hexa-hedra. Our method is based on the edge graph, which is defined by the sharp edges between the boundary surfaces of the solid. We repeatedly decompose the solid into smaller solids until all of them belong to a certain class of predefined base solids. The splitting step of the algorithm is based on simple combinatorial and geometric criteria. The segmentation technique described in the paper is part of a process pipeline for solving the isogeometric segmentation problem that we outline in the paper.
We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analy... more We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analysis (IGA). The input into our pipeline is a boundary represented 3D model, either as a triangulation or as a collection of trimmed non-uniform rational B-spline (NURBS) surfaces. The pipeline consists of three stages: computer aided design (CAD) model reconstruction from a triangulation (if necessary); segmentation of the boundary-represented solid into topological hexahe-dra; and volume parameterization. The result is a collection of volumetric NURBS patches. In this paper we discuss our methods for the three stages, and demonstrate the suitability of the result for IGA by performing stress simulations with examples of the output.
Purpose-The waste recycling industry increasingly relies on magnetic density separators. These de... more Purpose-The waste recycling industry increasingly relies on magnetic density separators. These devices generate an upward magnetic force in ferro-fluids allowing to separate the immersed particles according to their mass density. Recently, a new separator design has been proposed that significantly reduces the required amount of permanent magnet material. The purpose of this paper is to reduce the undesired end-effects in the upward force that this design generates by altering the shape of the fer-romagnetic covers of the individual poles. Design/methodology/approach-We represent the shape of the fer-romagnetic pole covers with B-splines and define a cost functional that measures the non-uniformity in the magnetic force in an area above the poles. We apply an isogeometric shape optimization procedure, which allows us to accurately represent, analyze and optimize the geometry using only a few design variables. The design problem is regularized by imposing constraints that enforce the convexity of the pole cover shapes. It is solved by a non-linear optimization procedure. We validate the implementation of our algorithm using a simplified variant of our design problem with a known analytical solution. The algorithm is subsequently applied to the problem posed. Research limitations/implications-The shape optimization attains its target and yields pole cover shapes that give rise to a magnetic field that is uniform over a larger domain. This increased uniformity is obtained at the cost of a pole cover shape that differs per pole. This limitation has negligible impact on the manufacturing of the separator. The new pole cover shapes, therefore, lead to improved performance of the density separation. Originality/value-This paper treats the shapes optimization of magnetic density separators systematically and presents new shapes for the ferromagnetic pole covers. Due to the larger uniformity of the generated field, these shapes should enable larger amounts of waste to be processed than the previous design.
In this thesis a recently proposed numerical method for solving partial differential equations,
i... more In this thesis a recently proposed numerical method for solving partial differential equations, isogeometric analysis (IGA), is utilized for the purpose of shape optimization, with a particular emphasis on applications to two-dimensional design problems arising in electromagnetic applications. The study is motivated by the fact that in contrast with most commonly utilized finite element approximations, IGA allows one to exactly represent geometries arising in computer aided design applications with relatively few variables using splines. The following problems coming from theoretical considerations or engineering applications are solved in the thesis utilizing IGA: finding a shape having a few prescribed eigenvalues of the Laplace operator; shape optimization of sub-wavelength micro-antennas for energy concentration; shape optimization of nano-antennas for field enhancement; economical design of magnetic density separators. From the point of view of method development, several heuristic approaches for extending a valid parametrization of the boundary onto the domain’s interior are examined in the thesis. The parametrization approaches and a method for validating a spline parametrization are combined into an iterative algorithm for shape optimization of two dimensional electromagnetic problems. The algorithm may also be relevant for problems in other engineering disciplines. Using the methods developed in this thesis, remarkably we have obtained antennas that perform one million times better than an earlier topology optimization result. This shows a great potential of shape optimization using IGA in the area of electromagnetic antenna design in particular, and for electromagnetic problems in general. Our conclusion is that IGA is well suited for shape optimization.
—We consider the benchmark problem of magnetic energy density enhancement in a small spatial regi... more —We consider the benchmark problem of magnetic energy density enhancement in a small spatial region by varying the shape of two symmetric conducting scatterers. We view this problem as a prototype for a wide variety of geometric design problems in electromagnetic applications. Our approach for solving this problem is based on shape optimization and isogeometric analysis. One of the major difficulties we face to make these methods work together is the need to maintain a valid parametrization of the computational domain during the optimization. Our approach to generating a domain parametrization is based on minimizing a second order approximation to the non-linear Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coefficients of a B-spline expansion of the Jacobian. The shape found by this approach outperforms earlier design computed using topology optimization by a factor of one billion.
We look at some succesfull examples of shape optimisation using isogeometric analysis. We also ad... more We look at some succesfull examples of shape optimisation using isogeometric analysis. We also addresses some problems which we encountered.
We consider a model problem of isogeometric shape optimization of vibrating membranes whose shape... more We consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely. The main obstacle we face is the need for robust and inexpensive extension of a B-spline parametrization from the boundary of a domain onto its interior, a task which has to be performed in every optimization iteration. We experiment with two numerical methods (one is based on the idea of constructing a quasi-conformal mapping, whereas the other is based on a spring-based mesh model) for carrying out this task, which turn out to work sufficiently well in the present situation. We perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth, optimized membrane shapes. Our conclusion is that isogeometric analysis fits well with shape optimization.
Motivated by the discretization problem in isogeometric analysis, we consider the challenge of se... more Motivated by the discretization problem in isogeometric analysis, we consider the challenge of segmenting a contractible boundary-represented solid into a small number of topological hexahedra. A satisfactory segmentation of a solid must eliminate non-convex edges because they prevent regular parameterizations. Our method works by searching a sufficiently connected edge graph of the solid for a cycle of vertices, called a cutting loop, which can be used to decompose the solid into two new solids with fewer non-convex edges. This can require the addition of auxiliary vertices to the edge graph. We provide theoretical justification for our approach by characterizing the cutting loops that can be used to segment the solid, and proving that the algorithm terminates. We select the cutting loop using a cost function. For this cost function we propose terms which help to select geometrically and combinatorially favorable cutting loops. We demonstrate the effects of these terms using a suite of examples.
In the context of segmenting a boundary represented solid into topological hexahedra suitable for... more In the context of segmenting a boundary represented solid into topological hexahedra suitable for isogeometric analysis, it is often necessary to split an existing face by constructing auxiliary curves. We consider solids represented as a collection of trimmed spline surfaces, and design a curve which can split the domain of a trimmed surface into two pieces satisfying the following criteria: the curve must not intersect the boundary of the original domain, it must not intersect itself, the two resulting pieces should have good shape, and the endpoints and the tangents of the curve at the endpoints must be equal to specified values.
In the context of segmenting a boundary represented solid into topological hexahedra suitable for... more In the context of segmenting a boundary represented solid into topological hexahedra suitable for isogeometric analysis, it is often necessary to split an existing face by constructing auxiliary curves. We consider solids represented as a collection of trimmed spline surfaces, and design a curve which can split the domain of a trimmed surface into two pieces satisfying the following criteria: the curve must not intersect the boundary of the original domain, it must not intersect itself, the two resulting pieces should have good shape, and the endpoints and the tangents of the curve at the endpoints must be equal to specified values.
We consider the generalized symmetric eigen-value problem where matrices depend smoothly on a par... more We consider the generalized symmetric eigen-value problem where matrices depend smoothly on a parameter. It is well known that in general individual eigen-values, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric polynomials of a number of eigen-values, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one.
Before isogeometric analysis can be applied to solving a partial differential equation posed over... more Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis-oriented methods. Their performance is illustrated through a few numerical examples.
ABSTRACT The problem of designing vibrating membranes was first considered by J.W. Hutchinson and... more ABSTRACT The problem of designing vibrating membranes was first considered by J.W. Hutchinson and F.I. Niordson [1]. In particular, they considered the problem of the design of a harmonic drum, i.e., a mem-brane where the eigenvalues of the Laplace operator are in harmonic proportion. This is related to M. Kac's famous question "Can one hear the shape of a drum?" [2]. Later, C. Kane and M. Schoenauer have attacked the problem by genetic algorithms [3] while the present work uses a systemetic gradient driven approach. We propose to use isogeometric analysis [4] and shape optimization to solve the problem. More precisely, we want to find the shape of a membrane whose first few eigenvalues are prescribed. As the eigenvalues do not determine the shape of the membrane (see [5]) we minimize the length of the perimeter under constraints on the first few eigenvalues. In the first example we design a membrane whose first four eigenvalues are λ 1 = 5.0122, λ 2 = 11.6349, λ 3 = 13.4102, λ 4 = 20.6025, and λ 5 = 23.6877, see figure 1 for the result. Figure 1: A "pear-shaped" region obtained by isoparametric design. Left: initial shape. Middle: opti-mized shape with parameter lines (red and cyan lines) and control points (blue and black dots) of the B-spline parametrization. Right: Jacobians of the parametrization at points of the uniform 500 × 500 mesh in which there are no non-positive Jacobians. The number of degrees of freedom of this design is DOFs = 256.
Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improv... more Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the top-of-the-line one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, convolution with PSIAC filters reduces to a stable inner product with the DG output and hence provides a stable and efficient implementation. The paper focuses on the remarkable fact that, for the canonical linear or non-linear hyperbolic test equation, new piecewise constant PSIAC filters of small support outperform the top-of-the-line boundary filters in the literature. These least-degree filters reduce the error at the boundaries to less than the error even of the symmetric filters of the same support. Due to their simplicity, and since this least degree filter has an exact symbolic form, convolution is stable as well as efficient and derivatives of the convolved output are easy to compute.
We present an edge graph segmentation for solids with convex and non-convex edges.
◮ Based on co... more We present an edge graph segmentation for solids with convex and non-convex edges. ◮ Based on combinatorial and geometric criteria. ◮ Convex edges: An implemented algorithm has been presented. ◮ Non-convex edges: First promising results → ongoing work.
Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improv... more Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the top-of-the-line one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, convolution with PSIAC filters reduces to a stable inner product with the DG output and hence provides a stable and efficient implementation. The paper focuses on the remarkable fact that, for the canonical linear or non-linear hyperbolic test equation, new piecewise constant PSIAC filters of small support outperform the top-of-the-line boundary filters in the literature. These least-degree filters reduce the error at the boundaries to less than the error even of the symmetric filters of the same support. Due to their simplicity, and since this least degree filter has an exact symbolic form, convolution is stable as well as efficient and derivatives of the convolved output are easy to compute.
Convolving the output of Discontinuous Galerkin computations with symmetric Smoothness-Increasing... more Convolving the output of Discontinuous Galerkin computations with symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. This paper interprets these filters as instances of a general class of position-dependent (PSIAC) spline filters that can have non-uniform knot sequences and skip B-splines of the sequence. PSIAC filters with rational knot sequences have rational coefficients. For prototype knot sequences , such as integer sequences that may have repeated entries, PSIAC filters can be expressed in symbolic form. Based on the insight that filters for shifted or scaled knot sequences are easily derived by non-uniform scaling of one prototype filter, a single filter can be re-used in different locations and at different scales. Computing a value of the convolution then simplifies to forming a scalar product of a short vector with the local output data. Restating one-sided filters in this form improves both stability and efficiency compared to their original formulation via numerical integration. PSIAC filtering is demonstrated for several established and one new boundary filter.
We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected e... more We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected edge graph consisting of only convex edges into a collection of topological hexa-hedra. Our method is based on the edge graph, which is defined by the sharp edges between the boundary surfaces of the solid. We repeatedly decompose the solid into smaller solids until all of them belong to a certain class of predefined base solids. The splitting step of the algorithm is based on simple combinatorial and geometric criteria. The segmentation technique described in the paper is part of a process pipeline for solving the isogeometric segmentation problem that we outline in the paper.
We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analy... more We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analysis (IGA). The input into our pipeline is a boundary represented 3D model, either as a triangulation or as a collection of trimmed non-uniform rational B-spline (NURBS) surfaces. The pipeline consists of three stages: computer aided design (CAD) model reconstruction from a triangulation (if necessary); segmentation of the boundary-represented solid into topological hexahe-dra; and volume parameterization. The result is a collection of volumetric NURBS patches. In this paper we discuss our methods for the three stages, and demonstrate the suitability of the result for IGA by performing stress simulations with examples of the output.
Purpose-The waste recycling industry increasingly relies on magnetic density separators. These de... more Purpose-The waste recycling industry increasingly relies on magnetic density separators. These devices generate an upward magnetic force in ferro-fluids allowing to separate the immersed particles according to their mass density. Recently, a new separator design has been proposed that significantly reduces the required amount of permanent magnet material. The purpose of this paper is to reduce the undesired end-effects in the upward force that this design generates by altering the shape of the fer-romagnetic covers of the individual poles. Design/methodology/approach-We represent the shape of the fer-romagnetic pole covers with B-splines and define a cost functional that measures the non-uniformity in the magnetic force in an area above the poles. We apply an isogeometric shape optimization procedure, which allows us to accurately represent, analyze and optimize the geometry using only a few design variables. The design problem is regularized by imposing constraints that enforce the convexity of the pole cover shapes. It is solved by a non-linear optimization procedure. We validate the implementation of our algorithm using a simplified variant of our design problem with a known analytical solution. The algorithm is subsequently applied to the problem posed. Research limitations/implications-The shape optimization attains its target and yields pole cover shapes that give rise to a magnetic field that is uniform over a larger domain. This increased uniformity is obtained at the cost of a pole cover shape that differs per pole. This limitation has negligible impact on the manufacturing of the separator. The new pole cover shapes, therefore, lead to improved performance of the density separation. Originality/value-This paper treats the shapes optimization of magnetic density separators systematically and presents new shapes for the ferromagnetic pole covers. Due to the larger uniformity of the generated field, these shapes should enable larger amounts of waste to be processed than the previous design.
In this thesis a recently proposed numerical method for solving partial differential equations,
i... more In this thesis a recently proposed numerical method for solving partial differential equations, isogeometric analysis (IGA), is utilized for the purpose of shape optimization, with a particular emphasis on applications to two-dimensional design problems arising in electromagnetic applications. The study is motivated by the fact that in contrast with most commonly utilized finite element approximations, IGA allows one to exactly represent geometries arising in computer aided design applications with relatively few variables using splines. The following problems coming from theoretical considerations or engineering applications are solved in the thesis utilizing IGA: finding a shape having a few prescribed eigenvalues of the Laplace operator; shape optimization of sub-wavelength micro-antennas for energy concentration; shape optimization of nano-antennas for field enhancement; economical design of magnetic density separators. From the point of view of method development, several heuristic approaches for extending a valid parametrization of the boundary onto the domain’s interior are examined in the thesis. The parametrization approaches and a method for validating a spline parametrization are combined into an iterative algorithm for shape optimization of two dimensional electromagnetic problems. The algorithm may also be relevant for problems in other engineering disciplines. Using the methods developed in this thesis, remarkably we have obtained antennas that perform one million times better than an earlier topology optimization result. This shows a great potential of shape optimization using IGA in the area of electromagnetic antenna design in particular, and for electromagnetic problems in general. Our conclusion is that IGA is well suited for shape optimization.
—We consider the benchmark problem of magnetic energy density enhancement in a small spatial regi... more —We consider the benchmark problem of magnetic energy density enhancement in a small spatial region by varying the shape of two symmetric conducting scatterers. We view this problem as a prototype for a wide variety of geometric design problems in electromagnetic applications. Our approach for solving this problem is based on shape optimization and isogeometric analysis. One of the major difficulties we face to make these methods work together is the need to maintain a valid parametrization of the computational domain during the optimization. Our approach to generating a domain parametrization is based on minimizing a second order approximation to the non-linear Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coefficients of a B-spline expansion of the Jacobian. The shape found by this approach outperforms earlier design computed using topology optimization by a factor of one billion.
We look at some succesfull examples of shape optimisation using isogeometric analysis. We also ad... more We look at some succesfull examples of shape optimisation using isogeometric analysis. We also addresses some problems which we encountered.
We consider a model problem of isogeometric shape optimization of vibrating membranes whose shape... more We consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely. The main obstacle we face is the need for robust and inexpensive extension of a B-spline parametrization from the boundary of a domain onto its interior, a task which has to be performed in every optimization iteration. We experiment with two numerical methods (one is based on the idea of constructing a quasi-conformal mapping, whereas the other is based on a spring-based mesh model) for carrying out this task, which turn out to work sufficiently well in the present situation. We perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth, optimized membrane shapes. Our conclusion is that isogeometric analysis fits well with shape optimization.
Motivated by the discretization problem in isogeometric analysis, we consider the challenge of se... more Motivated by the discretization problem in isogeometric analysis, we consider the challenge of segmenting a contractible boundary-represented solid into a small number of topological hexahedra. A satisfactory segmentation of a solid must eliminate non-convex edges because they prevent regular parameterizations. Our method works by searching a sufficiently connected edge graph of the solid for a cycle of vertices, called a cutting loop, which can be used to decompose the solid into two new solids with fewer non-convex edges. This can require the addition of auxiliary vertices to the edge graph. We provide theoretical justification for our approach by characterizing the cutting loops that can be used to segment the solid, and proving that the algorithm terminates. We select the cutting loop using a cost function. For this cost function we propose terms which help to select geometrically and combinatorially favorable cutting loops. We demonstrate the effects of these terms using a suite of examples.
In the context of segmenting a boundary represented solid into topological hexahedra suitable for... more In the context of segmenting a boundary represented solid into topological hexahedra suitable for isogeometric analysis, it is often necessary to split an existing face by constructing auxiliary curves. We consider solids represented as a collection of trimmed spline surfaces, and design a curve which can split the domain of a trimmed surface into two pieces satisfying the following criteria: the curve must not intersect the boundary of the original domain, it must not intersect itself, the two resulting pieces should have good shape, and the endpoints and the tangents of the curve at the endpoints must be equal to specified values.
In the context of segmenting a boundary represented solid into topological hexahedra suitable for... more In the context of segmenting a boundary represented solid into topological hexahedra suitable for isogeometric analysis, it is often necessary to split an existing face by constructing auxiliary curves. We consider solids represented as a collection of trimmed spline surfaces, and design a curve which can split the domain of a trimmed surface into two pieces satisfying the following criteria: the curve must not intersect the boundary of the original domain, it must not intersect itself, the two resulting pieces should have good shape, and the endpoints and the tangents of the curve at the endpoints must be equal to specified values.
We consider the generalized symmetric eigen-value problem where matrices depend smoothly on a par... more We consider the generalized symmetric eigen-value problem where matrices depend smoothly on a parameter. It is well known that in general individual eigen-values, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric polynomials of a number of eigen-values, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one.
Before isogeometric analysis can be applied to solving a partial differential equation posed over... more Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis-oriented methods. Their performance is illustrated through a few numerical examples.
ABSTRACT The problem of designing vibrating membranes was first considered by J.W. Hutchinson and... more ABSTRACT The problem of designing vibrating membranes was first considered by J.W. Hutchinson and F.I. Niordson [1]. In particular, they considered the problem of the design of a harmonic drum, i.e., a mem-brane where the eigenvalues of the Laplace operator are in harmonic proportion. This is related to M. Kac's famous question "Can one hear the shape of a drum?" [2]. Later, C. Kane and M. Schoenauer have attacked the problem by genetic algorithms [3] while the present work uses a systemetic gradient driven approach. We propose to use isogeometric analysis [4] and shape optimization to solve the problem. More precisely, we want to find the shape of a membrane whose first few eigenvalues are prescribed. As the eigenvalues do not determine the shape of the membrane (see [5]) we minimize the length of the perimeter under constraints on the first few eigenvalues. In the first example we design a membrane whose first four eigenvalues are λ 1 = 5.0122, λ 2 = 11.6349, λ 3 = 13.4102, λ 4 = 20.6025, and λ 5 = 23.6877, see figure 1 for the result. Figure 1: A "pear-shaped" region obtained by isoparametric design. Left: initial shape. Middle: opti-mized shape with parameter lines (red and cyan lines) and control points (blue and black dots) of the B-spline parametrization. Right: Jacobians of the parametrization at points of the uniform 500 × 500 mesh in which there are no non-positive Jacobians. The number of degrees of freedom of this design is DOFs = 256.
Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improv... more Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the top-of-the-line one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, convolution with PSIAC filters reduces to a stable inner product with the DG output and hence provides a stable and efficient implementation. The paper focuses on the remarkable fact that, for the canonical linear or non-linear hyperbolic test equation, new piecewise constant PSIAC filters of small support outperform the top-of-the-line boundary filters in the literature. These least-degree filters reduce the error at the boundaries to less than the error even of the symmetric filters of the same support. Due to their simplicity, and since this least degree filter has an exact symbolic form, convolution is stable as well as efficient and derivatives of the convolved output are easy to compute.
We present an edge graph segmentation for solids with convex and non-convex edges.
◮ Based on co... more We present an edge graph segmentation for solids with convex and non-convex edges. ◮ Based on combinatorial and geometric criteria. ◮ Convex edges: An implemented algorithm has been presented. ◮ Non-convex edges: First promising results → ongoing work.
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Papers by Manh Nguyen
isogeometric analysis (IGA), is utilized for the purpose of shape optimization, with a particular
emphasis on applications to two-dimensional design problems arising in electromagnetic applications.
The study is motivated by the fact that in contrast with most commonly utilized finite
element approximations, IGA allows one to exactly represent geometries arising in computer
aided design applications with relatively few variables using splines.
The following problems coming from theoretical considerations or engineering applications
are solved in the thesis utilizing IGA:
finding a shape having a few prescribed eigenvalues of the Laplace operator;
shape optimization of sub-wavelength micro-antennas for energy concentration;
shape optimization of nano-antennas for field enhancement;
economical design of magnetic density separators.
From the point of view of method development, several heuristic approaches for extending a
valid parametrization of the boundary onto the domain’s interior are examined in the thesis. The
parametrization approaches and a method for validating a spline parametrization are combined
into an iterative algorithm for shape optimization of two dimensional electromagnetic problems.
The algorithm may also be relevant for problems in other engineering disciplines.
Using the methods developed in this thesis, remarkably we have obtained antennas that
perform one million times better than an earlier topology optimization result. This shows a
great potential of shape optimization using IGA in the area of electromagnetic antenna design
in particular, and for electromagnetic problems in general. Our conclusion is that IGA is well
suited for shape optimization.
Drafts by Manh Nguyen
Conference Presentations by Manh Nguyen
◮ Based on combinatorial and geometric criteria.
◮ Convex edges: An implemented algorithm has been presented.
◮ Non-convex edges: First promising results → ongoing work.
isogeometric analysis (IGA), is utilized for the purpose of shape optimization, with a particular
emphasis on applications to two-dimensional design problems arising in electromagnetic applications.
The study is motivated by the fact that in contrast with most commonly utilized finite
element approximations, IGA allows one to exactly represent geometries arising in computer
aided design applications with relatively few variables using splines.
The following problems coming from theoretical considerations or engineering applications
are solved in the thesis utilizing IGA:
finding a shape having a few prescribed eigenvalues of the Laplace operator;
shape optimization of sub-wavelength micro-antennas for energy concentration;
shape optimization of nano-antennas for field enhancement;
economical design of magnetic density separators.
From the point of view of method development, several heuristic approaches for extending a
valid parametrization of the boundary onto the domain’s interior are examined in the thesis. The
parametrization approaches and a method for validating a spline parametrization are combined
into an iterative algorithm for shape optimization of two dimensional electromagnetic problems.
The algorithm may also be relevant for problems in other engineering disciplines.
Using the methods developed in this thesis, remarkably we have obtained antennas that
perform one million times better than an earlier topology optimization result. This shows a
great potential of shape optimization using IGA in the area of electromagnetic antenna design
in particular, and for electromagnetic problems in general. Our conclusion is that IGA is well
suited for shape optimization.
◮ Based on combinatorial and geometric criteria.
◮ Convex edges: An implemented algorithm has been presented.
◮ Non-convex edges: First promising results → ongoing work.