Computation of the Euler Number of a Binary Image Composed of Hexagonal Cells

ABSTRACTMost of the proposals to compute the Euler number of a binary image have been designed to work with imagescomposed of squared cells. Only a few of these methods (in the case of images composed of hexagonal cells) havebeen reported in literature, although it is known that images composed of hexagonal cells do not suffer from theproblems of connectivity frequently found in the case of images composed of squared cells. In this paper, a new wayto compute the Euler number (E) of a binary image composed of hexagonal cells is presented. For this, the perimeterP of the isolated regions in the image, their contact perimeter c P and the type T of a cell are used to obtain thisimportant invariant. The proposal can be used alone or in combination with other features to describe any binaryplanar shape composed of hexagonal pixels for its further recognition

Automated CAD Model Generation for Structural Optimisation

Computer-aided design (CAD) models play a crucial role in the design, manufacturing and maintenance of products. Therefore, the mesh-based finite element descriptions common in structural optimisation must be first translated into CAD models. Currently, this translation either can be performed semi-manually or fails to reserve the structural optimality found by the structural optimisation due to the intrinsic difference in geometric representation between finite element mesh and CAD model. This thesis propose a fully automated and topologically accurate approach to synthesise structurally sound parametric CAD models from topology-optimised finite element models to fill the long-existing gap between structural optimisation and CAD systems. This approach successfully preserves the optimal structural performance during the mesh-CAD conversion. The solution provided in this thesis is to first convert the topology-optimised structure into a spatial frame structure and then to regenerate it in a CAD system using standard constructive solid geometry (CSG) operations. The obtained parametric CAD models are compact, that is, have as few as possible geometric parameters, which makes them ideal for editing and further processing within a CAD system. The critical task of converting the topology-optimised structure into an optimal spatial frame structure is accomplished in several steps. The first step is to generate a one-voxel-wide voxel chain model from the topology-optimised voxel model using a topology-preserving skeletonisation algorithm from digital topology. The undirected graph defined by the voxel chain model yields a spatial frame structure after processing it with the proposed graph algorithms. Subsequently, the cross-sections and layout of the frame members are optimised to recover its optimality, which may have been compromised during the conversion process. At last, the obtained frame structure is generated in a CAD system by repeatedly combining primitive solids, like cylinders and spheres, using boolean operations. The resulting solid model is a boundary representation (B-Rep) consisting of trimmed non-uniform rational B-spline (NURBS) curves and surfaces. The numerical studies in this thesis clarify that the converted spatial frame structures are with equivalent structural performance. Moreover, CAD models generated from the spatial frame structures have significantly fewer geometric degree of freedom compared to the topology-optimised structures. Though the numerical studies use topology-optimised structures as input and compact CSG models as output, this thesis also provides the way to extend the proposed generation process to taking other optimised meshes and producing outputs of various geometric representations. This offers a wide range of possible applications and brings new thoughts to industrial design and manufacturing.Chinese Scholarship Counci

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

Topologically robust CAD model generation for structural optimisation

Computer-aided design (CAD) models play a crucial role in the design, manufacturing and maintenance of products. Therefore, the mesh-based finite element descriptions common in structural optimisation must be first translated into CAD models. Currently, this can at best be performed semi-manually. We propose a fully automated and topologically accurate approach to synthesise a structurally-sound parametric CAD model from topology optimised finite element models. Our solution is to first convert the topology optimised structure into a spatial frame structure and then to regenerate it in a CAD system using standard constructive solid geometry (CSG) operations. The obtained parametric CAD models are compact, that is, have as few as possible geometric parameters, which makes them ideal for editing and further processing within a CAD system. The critical task of converting the topology optimised structure into an optimal spatial frame structure is accomplished in several steps. We first generate from the topology optimised voxel model a one-voxel-wide voxel chain model using a topology-preserving skeletonisation algorithm from digital topology. The weighted undirected graph defined by the voxel chain model yields a spatial frame structure after processing it with standard graph algorithms. Subsequently, we optimise the cross-sections and layout of the frame members to recover its optimality, which may have been compromised during the conversion process. At last, we generate the obtained frame structure in a CAD system by repeatedly combining primitive solids, like cylinders and spheres, using boolean operations. The resulting solid model is a boundary representation (B-Rep) consisting of trimmed non-uniform rational B-spline (NURBS) curves and surfaces

COMPUTATION OF MINKOWSKI MEASURES ON 2D AND 3D BINARY IMAGES

Minkowski functionals encompass standard geometric parameters such as volume, area, length and the Euler-Poincaré characteristic. Software tools for computing approximations of Minkowski functionals on binary 2D or 3D images are now available based on mathematical methods due to Serra, Lang and Ohser. Minkowski functionals can not be used to describe spatial heterogeneity of structures. This description can be performed by using Minkowski measures, which are local versions of Minkowski functionals. In this paper, we discuss how to extend the computation of Minkowski functionals to the computation of Minkowski measures. Approximations of Minkowski measures are computed using fltering and look-up table transformations. The final result is represented as a grey-level image. Approximation errors are investigated based on numerical examples. Convergence and non convergence of the measure approximations are discussed. The measure of surface area is used to describe spatial heterogeneity of a synthetic structure, and of an image of tomato pericarp

Advanced operator splitting based semi-implicit spectral method to solve the binary and single component phase-field crystal model

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We present extensive testing in order to find the optimum balance among errors associated with time integration, spatial discretization, and splitting for a fully spectral semi implicit scheme of the phase field crystal model. The scheme solves numerically the equations of dissipative dynamics of the binary phase field crystal model proposed by Elder et al. [Elder et al, 2007]. The fully spectral semi implicit scheme uses the operator splitting method in order to decompose the complex equations in the phase field crystal model into sub-problems that can be solved more efficiently. Using the combination of non-trivial splitting with the spectral approach, the scheme leads to a set of algebraic equations of diagonal matrix form and thus easier to solve. Using this method developed by the BCAST research team we are able to show that it speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Comparing both the finite difference scheme used by Elder et al [Elder et al, 2007] to the spectral semi implicit scheme, we are also able to show that the finite differencing cannot compete with the spectral differencing in regards to accuracy. This is mainly due to numerical dissipation in finite differencing. In addition the results show that this method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs. We have applied the semi-implicit spectral scheme for binary alloys to explore polycrystalline dendritic solidification. The kinetics of transformation has been analysed in terms of Johnson-Mehl-Avrami-Kolmogorov formalism. We show that Avrami plots are not linear, and the respective Avrami-Kolmogorov exponents (PAK) vary with the transformed fraction (or time). Using the semi-implicit spectral scheme we have been able to provide extensive numerical testing of methods in solving the single component case. This has been demonstrated by using unconditional time stepping with comparable simulations using conditional time stepping. We show the accuracy of the solution for unconditional time stepping is not compromised and furthermore computational efficiency can be significantly increased with the introduction of this scheme. Finally we have investigated how the composition of the initial liquid phase influences the eutectic morphology evolving during solidification. This is the first study that addresses this question using the dynamical density functional theory.EPSR

Geometric data understanding : deriving case specific features

There exists a tradition using precise geometric modeling, where uncertainties in data can be considered noise. Another tradition relies on statistical nature of vast quantity of data, where geometric regularity is intrinsic to data and statistical models usually grasp this level only indirectly. This work focuses on point cloud data of natural resources and the silhouette recognition from video input as two real world examples of problems having geometric content which is intangible at the raw data presentation. This content could be discovered and modeled to some degree by such machine learning (ML) approaches like deep learning, but either a direct coverage of geometry in samples or addition of special geometry invariant layer is necessary. Geometric content is central when there is a need for direct observations of spatial variables, or one needs to gain a mapping to a geometrically consistent data representation, where e.g. outliers or noise can be easily discerned. In this thesis we consider transformation of original input data to a geometric feature space in two example problems. The first example is curvature of surfaces, which has met renewed interest since the introduction of ubiquitous point cloud data and the maturation of the discrete differential geometry. Curvature spectra can characterize a spatial sample rather well, and provide useful features for ML purposes. The second example involves projective methods used to video stereo-signal analysis in swimming analytics. The aim is to find meaningful local geometric representations for feature generation, which also facilitate additional analysis based on geometric understanding of the model. The features are associated directly to some geometric quantity, and this makes it easier to express the geometric constraints in a natural way, as shown in the thesis. Also, the visualization and further feature generation is much easier. Third, the approach provides sound baseline methods to more traditional ML approaches, e.g. neural network methods. Fourth, most of the ML methods can utilize the geometric features presented in this work as additional features.Geometriassa käytetään perinteisesti tarkkoja malleja, jolloin datassa esiintyvät epätarkkuudet edustavat melua. Toisessa perinteessä nojataan suuren datamäärän tilastolliseen luonteeseen, jolloin geometrinen säännönmukaisuus on datan sisäsyntyinen ominaisuus, joka hahmotetaan tilastollisilla malleilla ainoastaan epäsuorasti. Tämä työ keskittyy kahteen esimerkkiin: luonnonvaroja kuvaaviin pistepilviin ja videohahmontunnistukseen. Nämä ovat todellisia ongelmia, joissa geometrinen sisältö on tavoittamattomissa raakadatan tasolla. Tämä sisältö voitaisiin jossain määrin löytää ja mallintaa koneoppimisen keinoin, esim. syväoppimisen avulla, mutta joko geometria pitää kattaa suoraan näytteistämällä tai tarvitaan neuronien lisäkerros geometrisia invariansseja varten. Geometrinen sisältö on keskeinen, kun tarvitaan suoraa avaruudellisten suureiden havainnointia, tai kun tarvitaan kuvaus geometrisesti yhtenäiseen dataesitykseen, jossa poikkeavat näytteet tai melu voidaan helposti erottaa. Tässä työssä tarkastellaan datan muuntamista geometriseen piirreavaruuteen kahden esimerkkiohjelman suhteen. Ensimmäinen esimerkki on pintakaarevuus, joka on uudelleen virinneen kiinnostuksen kohde kaikkialle saatavissa olevan datan ja diskreetin geometrian kypsymisen takia. Kaarevuusspektrit voivat luonnehtia avaruudellista kohdetta melko hyvin ja tarjota koneoppimisessa hyödyllisiä piirteitä. Toinen esimerkki koskee projektiivisia menetelmiä käytettäessä stereovideosignaalia uinnin analytiikkaan. Tavoite on löytää merkityksellisiä paikallisen geometrian esityksiä, jotka samalla mahdollistavat muun geometrian ymmärrykseen perustuvan analyysin. Piirteet liittyvät suoraan johonkin geometriseen suureeseen, ja tämä helpottaa luonnollisella tavalla geometristen rajoitteiden käsittelyä, kuten väitöstyössä osoitetaan. Myös visualisointi ja lisäpiirteiden luonti muuttuu helpommaksi. Kolmanneksi, lähestymistapa suo selkeän vertailumenetelmän perinteisemmille koneoppimisen lähestymistavoille, esim. hermoverkkomenetelmille. Neljänneksi, useimmat koneoppimismenetelmät voivat hyödyntää tässä työssä esitettyjä geometrisia piirteitä lisäämällä ne muiden piirteiden joukkoon