1,182 research outputs found
From approximating to interpolatory non-stationary subdivision schemes with the same generation properties
In this paper we describe a general, computationally feasible strategy to
deduce a family of interpolatory non-stationary subdivision schemes from a
symmetric non-stationary, non-interpolatory one satisfying quite mild
assumptions. To achieve this result we extend our previous work [C.Conti,
L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to
full generality by removing additional assumptions on the input symbols. For
the so obtained interpolatory schemes we prove that they are capable of
reproducing the same exponential polynomial space as the one generated by the
original approximating scheme. Moreover, we specialize the computational
methods for the case of symbols obtained by shifted non-stationary affine
combinations of exponential B-splines, that are at the basis of most
non-stationary subdivision schemes. In this case we find that the associated
family of interpolatory symbols can be determined to satisfy a suitable set of
generalized interpolating conditions at the set of the zeros (with reversed
signs) of the input symbol. Finally, we discuss some computational examples by
showing that the proposed approach can yield novel smooth non-stationary
interpolatory subdivision schemes possessing very interesting reproduction
properties
Piecewise Extended Chebyshev Spaces: a numerical test for design
Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all
of the same dimension, we join them via convenient connection matrices without
increasing the dimension. The global space is called a Piecewise Extended
Chebyshev (PEC) Space. In such a space one can count the total number of zeroes
of any non-zero element, exactly as in each EC-section-space. When this number
is bounded above in the global space the same way as in its section-spaces, we
say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of
ECP-spaces has been developed in the last two decades in relation to blossoms,
with a view to design. In particular, extending a classical procedure for
EC-spaces, ECP-spaces were recently proved to all be obtained by means of
piecewise generalised derivatives. This yields an interesting constructive
characterisation of ECP-spaces. Unfortunately, except for low dimensions and
for very few adjacent intervals, this characterisation proved to be rather
difficult to handle in practice. To try to overcome this difficulty, in the
present article we show how to reinterpret the constructive characterisation as
a theoretical procedure to determine whether or not a given PEC-space is an
ECP-space. This procedure is then translated into a numerical test, whose
usefulness is illustrated by relevant examples
Neural-network-based curve fitting using totally positive rational bases
This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational Bézier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation
Curve Skeleton and Moments of Area Supported Beam Parametrization in Multi-Objective Compliance Structural Optimization
This work addresses the end-to-end virtual automation of structural optimization up to the derivation of a parametric geometry model that can be used for application areas such as additive manufacturing or the verification of the structural optimization result with the finite element method.
A holistic design in structural optimization can be achieved with the weighted sum method, which can be automatically parameterized with curve skeletonization and cross-section regression to virtually verify the result and control the local size for additive manufacturing.
is investigated in general. In this paper, a holistic design is understood as a design that considers various compliances as an objective function. This parameterization uses the automated determination of beam parameters by so-called curve skeletonization with subsequent cross-section shape parameter estimation based on moments of area, especially for multi-objective optimized shapes. An essential contribution is the linking of the parameterization with the results of the structural optimization, e.g., to include properties such as boundary conditions, load conditions, sensitivities or even density variables in the curve skeleton parameterization.
The parameterization focuses on guiding the skeletonization based on the information provided by the optimization and the finite element model. In addition, the cross-section detection considers circular, elliptical, and tensor product spline cross-sections that can be applied to various shape descriptors such as convolutional surfaces, subdivision surfaces, or constructive solid geometry. The shape parameters of these cross-sections are estimated using stiffness distributions, moments of area of 2D images, and convolutional neural networks with a tailored loss function to moments of area. Each final geometry is designed by extruding the cross-section along the appropriate curve segment of the beam and joining it to other beams by using only unification operations.
The focus of multi-objective structural optimization considering 1D, 2D and 3D elements is on cases that can be modeled using equations by the Poisson equation and linear elasticity. This enables the development of designs in application areas such as thermal conduction, electrostatics, magnetostatics, potential flow, linear elasticity and diffusion, which can be optimized in combination or individually. Due to the simplicity of the cases defined by the Poisson equation, no experts are required, so that many conceptual designs can be generated and reconstructed by ordinary users with little effort.
Specifically for 1D elements, a element stiffness matrices for tensor product spline cross-sections are derived, which can be used to optimize a variety of lattice structures and automatically convert them into free-form surfaces. For 2D elements, non-local trigonometric interpolation functions are used, which should significantly increase interpretability of the density distribution. To further improve the optimization, a parameter-free mesh deformation is embedded so that the compliances can be further reduced by locally shifting the node positions.
Finally, the proposed end-to-end optimization and parameterization is applied to verify a linear elasto-static optimization result for and to satisfy local size constraint for the manufacturing with selective laser melting of a heat transfer optimization result for a heat sink of a CPU. For the elasto-static case, the parameterization is adjusted until a certain criterion (displacement) is satisfied, while for the heat transfer case, the manufacturing constraints are satisfied by automatically changing the local size with the proposed parameterization. This heat sink is then manufactured without manual adjustment and experimentally validated to limit the temperature of a CPU to a certain level.:TABLE OF CONTENT III
I LIST OF ABBREVIATIONS V
II LIST OF SYMBOLS V
III LIST OF FIGURES XIII
IV LIST OF TABLES XVIII
1. INTRODUCTION 1
1.1 RESEARCH DESIGN AND MOTIVATION 6
1.2 RESEARCH THESES AND CHAPTER OVERVIEW 9
2. PRELIMINARIES OF TOPOLOGY OPTIMIZATION 12
2.1 MATERIAL INTERPOLATION 16
2.2 TOPOLOGY OPTIMIZATION WITH PARAMETER-FREE SHAPE OPTIMIZATION 17
2.3 MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION WITH THE WEIGHTED SUM METHOD 18
3. SIMULTANEOUS SIZE, TOPOLOGY AND PARAMETER-FREE SHAPE OPTIMIZATION OF WIREFRAMES WITH B-SPLINE CROSS-SECTIONS 21
3.1 FUNDAMENTALS IN WIREFRAME OPTIMIZATION 22
3.2 SIZE AND TOPOLOGY OPTIMIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 27
3.3 PARAMETER-FREE SHAPE OPTIMIZATION EMBEDDED IN SIZE OPTIMIZATION 32
3.4 WEIGHTED SUM SIZE AND TOPOLOGY OPTIMIZATION 36
3.5 CROSS-SECTION COMPARISON 39
4. NON-LOCAL TRIGONOMETRIC INTERPOLATION IN TOPOLOGY OPTIMIZATION 41
4.1 FUNDAMENTALS IN MATERIAL INTERPOLATIONS 43
4.2 NON-LOCAL TRIGONOMETRIC SHAPE FUNCTIONS 45
4.3 NON-LOCAL PARAMETER-FREE SHAPE OPTIMIZATION WITH TRIGONOMETRIC SHAPE FUNCTIONS 49
4.4 NON-LOCAL AND PARAMETER-FREE MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION 54
5. FUNDAMENTALS IN SKELETON GUIDED SHAPE PARAMETRIZATION IN TOPOLOGY OPTIMIZATION 58
5.1 SKELETONIZATION IN TOPOLOGY OPTIMIZATION 61
5.2 CROSS-SECTION RECOGNITION FOR IMAGES 66
5.3 SUBDIVISION SURFACES 67
5.4 CONVOLUTIONAL SURFACES WITH META BALL KERNEL 71
5.5 CONSTRUCTIVE SOLID GEOMETRY 73
6. CURVE SKELETON GUIDED BEAM PARAMETRIZATION OF TOPOLOGY OPTIMIZATION RESULTS 75
6.1 FUNDAMENTALS IN SKELETON SUPPORTED RECONSTRUCTION 76
6.2 SUBDIVISION SURFACE PARAMETRIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 78
6.3 CURVE SKELETONIZATION TAILORED TO TOPOLOGY OPTIMIZATION WITH PRE-PROCESSING 82
6.4 SURFACE RECONSTRUCTION USING LOCAL STIFFNESS DISTRIBUTION 86
7. CROSS-SECTION SHAPE PARAMETRIZATION FOR PERIODIC B-SPLINES 96
7.1 PRELIMINARIES IN B-SPLINE CONTROL GRID ESTIMATION 97
7.2 CROSS-SECTION EXTRACTION OF 2D IMAGES 101
7.3 TENSOR SPLINE PARAMETRIZATION WITH MOMENTS OF AREA 105
7.4 B-SPLINE PARAMETRIZATION WITH MOMENTS OF AREA GUIDED CONVOLUTIONAL NEURAL NETWORK 110
8. FULLY AUTOMATED COMPLIANCE OPTIMIZATION AND CURVE-SKELETON PARAMETRIZATION FOR A CPU HEAT SINK WITH SIZE CONTROL FOR SLM 115
8.1 AUTOMATED 1D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINED SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 118
8.2 AUTOMATED 2D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINT SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 120
8.3 USING THE HEAT SINK PROTOTYPES COOLING A CPU 123
9. CONCLUSION 127
10. OUTLOOK 131
LITERATURE 133
APPENDIX 147
A PREVIOUS STUDIES 147
B CROSS-SECTION PROPERTIES 149
C CASE STUDIES FOR THE CROSS-SECTION PARAMETRIZATION 155
D EXPERIMENTAL SETUP 15
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Geometric properties and constrained modification of trigonometric spline curves of Han
New types of quadratic and cubic trigonometrial polynomial curves have
been introduced in [2] and [3]. These trigonometric curves have a global shape
parameter λ. In this paper the geometric effect of this shape parameter on the
curves is discussed. We prove that this effect is linear. Moreover we show that
the quadratic curve can interpolate the control points at λ = √2. Constrained
modification of these curves is also studied. A curve passing through a given
point is computed by an algorithm which includes numerical computations.
These issues are generalized for surfaces with two shape parameters. We show
that a point of the surface can move along a hyperbolic paraboloid
Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory
In this paper, we present an effectively numerical approach based on
isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT)
for geometrically nonlinear analysis of laminated composite plates. The HSDT
allows us to approximate displacement field that ensures by itself the
realistic shear strain energy part without shear correction factors. IGA
utilizing basis functions namely B-splines or non-uniform rational B-splines
(NURBS) enables to satisfy easily the stringent continuity requirement of the
HSDT model without any additional variables. The nonlinearity of the plates is
formed in the total Lagrange approach based on the von-Karman strain
assumptions. Numerous numerical validations for the isotropic, orthotropic,
cross-ply and angle-ply laminated plates are provided to demonstrate the
effectiveness of the proposed method
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