The Construction of Conforming-to-shape Truss Lattice Structures via 3D Sphere Packing

Truss lattices are common in a wide variety of engineering applications, due to their high ratio of strength versus relative density. They are used both as the interior support for other structures, and as structures on their own. Using 3D sphere packing, we propose a set of methods for generating truss lattices that fill the interior of B-rep models, polygonal or (trimmed) NURBS based, of arbitrary shape. Once the packing of the spheres has been established, beams between the centers of adjacent spheres are constructed, as spline based B-rep geometry. We also demonstrate additional capabilities of our methods, including connecting the truss lattice to (a shell of) the B-rep model, as well as constructing a tensor-product trivariate volumetric representation of the truss lattice - an important step towards direct compatibility for analysis.RYC-2017-2264

Heterogeneous Parametric Trivariate Fillets

Blending and filleting are well established operations in solid modeling and computer-aided geometric design. The creation of a transition surface which smoothly connects the boundary surfaces of two (or more) objects has been extensively investigated. In this work, we introduce several algorithms for the construction of, possibly heterogeneous, trivariate fillets, that support smooth filleting operations between pairs of, possibly heterogeneous, input trivariates. Several construction methods are introduced that employ functional composition algorithms as well as introduce a half Volumetric Boolean sum operation. A volumetric fillet, consisting of one or more tensor product trivariate(s), is fitted to the boundary surfaces of the input. The result smoothly blends between the two inputs, both geometrically and material-wise (properties of arbitrary dimension). The application of encoding heterogeneous material information into the constructed fillet is discussed and examples of all proposed algorithms are presented

Methods for constraint-based conceptual free-form surface design

Zusammenfassung Der constraint-basierte Entwurf von Freiformfl„chen ist eine m„chtige Methode im Computer gest�tzten Entwurf. Bekannte Realisierungen beschr„nken sich jedoch meist auf Interpolation von Rand- und isoparametrischen Kurven. In diesem Zusammenhang sind die sog. "Multi-patch" Methoden die am weitesten verbreitete Vorgehensweise. Hier versucht man Fl„chenverb„nde aus einem Netz von dreidimensionalen Kurven (oft gemischt mit unstrukturierten Punktewolken) derart zu generieren, dass die Kurven und Punkte von den Fl„chen interpoliert werden. Die Kurven werden als R„nder von rechteckigen oder dreieckigen bi-polynomialen oder polynomialen Fl„chen betrachtet. Unter dieser Einschr„nkung leidet die Flexibilit„t des Verfahrens. In dieser Dissertation schlagen wir vor, beliebige, d.h. auch nicht iso-parametrische, Kurven zu verwenden. Dadurch ergeben sich folgende Vorteile: Erstens kann so beispielsweise eine B-spline Fl„che entlang einer benutzerdefinierten Kurve verformt werden w„hrend andere Kurven oder Punkte fixiert sind. Zweitens, kann eine B-spline Fl„che Kurven interpolieren, die sich nicht auf iso-parametrische Linien der Fl„che abbilden lassen. Wir behandeln drei Arten von Constraints: Inzidenz einer beliebigen Kurve auf einer B-spline Fl„che, Fixieren von Fl„chennormalen entlang einer beliebigen Kurve (dieser Constraint dient zur Herstellung von tangentialen šberg„ngen zwischen zwei Fl„chen) und die sog. Variational Constrains. Letztere dienen unter anderem zur Optimierung der physikalischen und optischen Eigenschaften der Fl„chen. Es handelt sich hierbei um die Gausschen Normalgleichungen der Fl„chenfunktionale zweiter Ordnung, wie sie in der Literatur bekannt sind. Die Dissertation gliedert sich in zwei Teile. Der erste Teil befasst sich mit der Aufstellung der linearen Gleichungssysteme, welche die oben erw„hnten Constraints repr„sentieren. Der zweite Teil behandelt Methoden zum L”sen dieser Gleichungssysteme. Der Kern des ersten Teiles ist die Erweiterung und Generalisierung des auf Polarformen (Blossoms) basierenden Algorithmus f�r Verkettung von Polynomen auf Bezier und B-spline Basis: Gegeben sei eine B-spline Fl„che und eine B-spline Kurve im Parameterraum der Fl„che. Wir zeigen, dass die Kontrollpunkte der dreidimensionalen Fl„chenkurve, welche als polynomiale Verkettung der beiden definiert ist, durch eine im Voraus berechenbare lineare Tranformation (eine Matrix) der Fl„chenkontrollpunkte ausgedr�ckt werden k”nnen. Dadurch k”nnen Inzidenzbeziehungen zwischen Kurven und Fl„chen exakt und auf eine sehr elegante und kompakte Art definiert werden. Im Vergleich zu den bekannten Methoden ist diese Vorgehensweise effizienter, numerisch stabiler und erh”ht nicht die Konditionszahl der zu l”senden linearen Gleichungen. Die Effizienz wird erreicht durch Verwendung von eigens daf�r entwickelten Datenstrukturen und sorgf„ltige Analyse von kombinatorischen Eigenschaften von Polarformen. Die Gleichungen zur Definition von Tangentialit„ts- und Variational Constraints werden als Anwendung und Erweiterung dieses Algorithmus implementiert. Beschrieben werden auch symbolische und numerische Operationen auf B-spline Polynomen (Multiplikation, Differenzierung, Integration). Dabei wird konsistent die Matrixdarstellung von B-spline Polynomen verwendet. Das L”sen dieser Art von Constraintproblemen bedeutet das Finden der Kontrollpunkte einer B-spline Fl„che derart, dass die definierten Bedingungen erf�llt werden. Dies wird durch L”sen von, im Allgemeinen, unterbestimmten und schlecht konditionierten linearen Gleichungssystemen bewerkstelligt. Da in solchen F„llen keine eindeutige, numerisch stabile L”sung existiert, f�hren die �blichen Methoden zum L”sen von linearen Gleichungssystemen nicht zum Erfolg. Wir greifen auf die Anwendung von sog. Regularisierungsmethoden zur�ck, die auf der Singul„rwertzerlegung (SVD) der Systemmatrix beruhen. Insbesondere wird die L-curve eingesetzt, ein "numerischer Hochfrequenzfilter", der uns in die Lage versetzt eine stabile L”sung zu berechnen. Allerdings reichen auch diese Methoden im Allgemeinen nicht aus, eine Fl„che zu generieren, welche die erw�nschten „sthetischen und physikalischen Eigenschaften besitzt. Verformt man eine Tensorproduktfl„che entlang einer nicht isoparametrischen Kurve, entstehen unerw�nschte Oszillationen und Verformungen. Dieser Effekt wird "Surface-Aliasing" genannt. Wir stellen zwei Methoden vor um diese Aliasing-Effekte zu beseitigen: Die erste Methode wird vorzugsweise f�r Deformationen einer existierenden B-spline Fl„che entlang einer nicht isoparametrischen Kurve angewendet. Es erfogt eine Umparametrisierung der zu verformenden Fl„che derart, dass die Kurve in der neuen Fl„che auf eine isoparametrische Linie abgebildet wird. Die Umparametrisierung einer B- spline Fl„che ist keine abgeschlossene Operation; die resultierende Fl„che besitzt i.A. keine B-spline Darstellung. Wir berechnen eine beliebig genaue Approximation der resultierenden Fl„che mittels Interpolation von Kurvennetzen, die von der umzuparametrisierenden Fl„che gewonnen werden. Die zweite Methode ist rein algebraisch: Es werden zus„tzliche Bedingungen an die L”sung des Gleichungssystems gestellt, die die Aliasing-Effekte unterdr�cken oder ganz beseitigen. Es wird ein restriktionsgebundenes Minimum einer Zielfunktion gesucht, deren globales Minimum bei "optimaler" Form der Fl„che eingenommen wird. Als Zielfunktionen werden Gl„ttungsfunktionale zweiter Ordnung eingesetzt. Die stabile L”sung eines solchen Optimierungsproblems kann aufgrund der nahezu linearen Abh„ngigkeit des Gleichungen nur mit Hilfe von Regularisierungsmethoden gewonnen werden, welche die vorgegebene Zielfunktion ber�cksichtigen. Wir wenden die sog. Modifizierte Singul„rwertzerlegung in Verbindung mit dem L-curve Filter an. Dieser Algorithmus minimiert den Fehler f�r die geometrischen Constraints so, dass die L”sung gleichzeitig m”glichst nah dem Optimum der Zielfunktion ist.The constrained-based design of free-form surfaces is currently limited to tensor-product interpolation of orthogonal curve networks or equally spaced grids of points. The, so- called, multi-patch methods applied mainly in the context of scattered data interpolation construct surfaces from given boundary curves and derivatives along them. The limitation to boundary curves or iso-parametric curves considerably lowers the flexibility of this approach. In this thesis, we propose to compute surfaces from arbitrary (that is, not only iso-parametric) curves. This allows us to deform a B-spline surface along an arbitrary user-defined curve, or, to interpolate a B-spline surface through a set of curves which cannot be mapped to iso-parametric lines of the surface. We consider three kinds of constraints: the incidence of a curve on a B-spline surface, prescribed surface normals along an arbitrary curve incident on a surface and the, so-called, variational constraints which enforce a physically and optically advantageous shape of the computed surfaces. The thesis is divided into two parts: in the first part, we describe efficient methods to set up the equations for above mentioned linear constraints between curves and surfaces. In the second part, we discuss methods for solving such constraints. The core of the first part is the extension and generalization of the blossom-based polynomial composition algorithm for B-splines: let be given a B-spline surface and a B-spline curve in the domain of that surface. We compute a matrix which represents a linear transformation of the surface control points such that after the transformation we obtain the control points of the curve representing the polynomial composition of the domain curve and the surface. The result is a 3D B-spline curve always exactly incident on the surface. This, so-called, composition matrix represents a set of linear curve-surface incidence constraints. Compared to methods used previously our approach is more efficient, numerically more stable and does not unnecessarily increase the condition number of the matrix. The thesis includes a careful analysis of the complexity and combinatorial properties of the algorithm. We also discuss topics regarding algebraic operations on B-spline polynomials (multiplication, differentiation, integration). The matrix representation of B-spline polynomials is used throughout the thesis. We show that the equations for tangency and variational constraints are easily obtained re-using the methods elaborated for incidence constraints. The solving of generalized curve-surface constraints means to find the control points of the unknown surface given one or several curves incident on that surface. This is accomplished by solving of large and, generally, under-determined and badly conditioned linear systems of equations. In such cases, no unique and numerically stable solution exists. Hence, the usual methods such as Gaussian elimination or QR-decomposition cannot be applied in straightforward manner. We propose to use regularization methods based on Singular Value Decomposition (SVD). We apply the so-called L-curve, which can be seen as an numerical high-frequency filter. The filter automatically singles out a stable solution such that best possible satisfaction of defined constraints is achieved. However, even the SVD along with the L-curve filter cannot be applied blindly: it turns out that it is not sufficient to require only algebraic stability of the solution. Tensor-product surfaces deformed along arbitrary incident curves exhibit unwanted deformations due to the rectangular structure of the model space. We discuss a geometric and an algebraic method to remove this, so-called, Surface aliasing effect. The first method reparametrizes the surface such that a general curve constraint is converted to iso-parametric curve constraint which can be easily solved by standard linear algebra methods without aliasing. The reparametrized surface is computed by means of the approximated surface-surface composition algorithm, which is also introduced in this thesis. While this is not possible symbolically, an arbitrary accurate approximation of the resulting surface is obtained using constrained curve network interpolation. The second method states additional constraints which suppress or completely remove the aliasing. Formally we solve a constrained least square approximation problem: we minimize an surface objective function subject to defined curve constraints. The objective function is chosen such that it takes in the minimal value if the surface has optimal shape; we use a linear combination of second order surface smoothing functionals. When solving such problems we have to deal with nearly linearly dependent equations. Problems of this type are called ill-posed. Therefore sophisticated numerical methods have to be applied in order to obtain a set of degrees of freedom (control points of the surface) which are sufficient to satisfy given constraints. The remaining unused degrees of freedom are used to enforce an optically pleasing shape of the surface. We apply the Modified Truncated SVD (MTSVD) algorithm in connection with the L-curve filter which determines a compromise between an optically pleasant shape of the surface and constraint satisfaction in a particularly efficient manner

Manufacturability and Analysis of Topologically Optimized Continuous Fiber Reinforced Composites

Researchers are unlocking the potential of Continuous Fiber Reinforced Composites for producing components with greater strength-to-weight ratios than state of the art metal alloys and unidirectional composites. The key is the emerging technology of topology optimization and advances in additive manufacturing. Topology optimization can fine tune component geometry and fiber placement all while satisfying stress constraints. However, the technology cannot yet robustly guarantee manufacturability. For this reason, substantial post-processing of an optimized design consisting of manual fiber replacement and subsequent Finite Element Analysis (FEA) is still required. To automate this post-processing in two dimensions, two (2) algorithms were developed. The first one is aimed at filling the space of a topologically optimized component with fibers of prescribed thickness. The objective is to produce flawless fiber paths, meaning no self-intersections, no tight turns, and no overlapping between fibers. It does so by leveraging concepts from elementary geometry and the Signed Distance Function of a topologically optimized domain. The manufacturable fiber paths are represented using Non-Uniform Rational Basis Splines, which can be readily conveyed to a 3D-printer as The second algorithm then calls a meshing routine to spatially discretize the topologically optimized domain. It takes input from the first algorithm to automatically create and append, orientations and material flags to the spatial elements produced by the meshing routine. Finally, it generates output that is then input to FEA software. The software is written in the C-programming language using the PETSc library. A load case is validated against MSC NASTRAN

Leveraging Relational Structure through Message Passing for Modelling Non-Euclidean Data

Modelling non-Euclidean data is difficult since objects for comparison can be formed of different numbers of constituent parts with different numbers of relations between them, and traditional (Euclidean) methods are non-trivial to apply. Message passing enables such modelling by leveraging the structure of the relations within a (or between) given object(s) in order to represent and compare structure in a vectorized form of fixed dimensions. In this work, we contribute novel message passing techniques that improve state of the art for non-Euclidean modelling in a set of specifically chosen domains. In particular, (1) we introduce an attention-based structure-aware global pooling operator for graph classification and demonstrate its effectiveness on a range of chemical property prediction benchmarks, we also show that our method outperforms state of the art graph classifiers in a graph isomorphism test, and demonstrate the interpretability of our method with respect to the learned attention coefficients. (2) We propose a style similarity measure for Boundary Representations (B-Reps) that leverages the style signals in the second order statistics of the activations in a pre-trained (unsupervised) 3D encoder, and learns their relative importance to an end-user through few-shot learning. Our approach differs from existing data-driven 3D style methods since it may be used in completely unsupervised settings. We show quantitatively that our proposed method with B-Reps is able to capture stronger style signals than alternative methods on meshes and point clouds despite its significantly greater computational efficiency. We also show it is able to generate meaningful style gradients with respect to the input shape. (3) We introduce a novel message passing-based model of computation and demonstrate its effectiveness in expressing the complex dependencies of biological systems necessary to model life-like systems and tracing cell lineage during cancerous tumour growth, and demonstrate the improvement over existing methods in terms of post-analysis

Filleting and Rounding using Trimmed Tensor Product Surfaces

The tensor product Bézier and NURBs surface representation is frequently exploited in computer aided geometric design. Yet, this representation is inherently rectangular, a topology that does not easily enable the skinning, filleting, and rounding of triangular regions or domains with arbitrary n-sided boundaries. Modern solid modeling systems support tensor product B'ezier and NURBs surfaces with the additional ability to represent the trimmed form of these surfaces. This paper explores and presents an approach that allows one to construct regular, nondegenerate positional or tangent plane continuous triangular or n-sided patches, each one as a trimmed tensor product surface. The proposed method is demonstrated on rounding of triangular corners using positional and tangent plane continuity conditions as well as an example of a C 0 hexagonal filleting patch

Morphing Waveriders for Atmospheric Entry

The primary challenge for vehicles entering planetary atmospheres is surviving the intense heating and deceleration encountered during the entry process. Entry capsules use sacrificial ablative heat shields and sustain several g deceleration. The high lift produced by the Space Shuttle geometry resulted in lower rates of heating and deceleration. This enabled a fully reusable vehicle that was protected by heat shield tiles. Hypersonic waveriders are vehicles that conform to the shape of the shock wave created by the vehicle. This produces high compression-lift and low drag, but only around a design Mach number. Atmospheric entry can reach speeds from zero to as high as Mach 40. A morphing waverider is a vehicle that deflects its flexible bottom surface as a function of Mach number in order to preserve a desired shock wave shape. It was demonstrated in this work that doing so retains high aerodynamic lift and lift-to-drag ratio across a wide range of Mach number. Numerical simulations were conducted for case-study waveriders designed for Mach 6 and 8 for flight at their design conditions as well as with variations in angle-of-attack and Mach number. A single-species air model was used between Mach 1 and 12 with the RANS k-omega SST and LES-WALE turbulence models. A seven-species air model was used for Mach 15 at 60km altitude and Mach 20 at 75km. Analytical methods were used to construct a reduced-order model (ROM) for estimating waverider aerodynamic forces, moments, and heating. The ROM matched numerical simulation results within 5-10% for morphing waveriders with variations in angle-of-attack, but discrepancies exceeded 20% for large deviations of rigid vehicles from their design Mach numbers. Atmospheric entry trajectory simulations were conducted using reduced-order models for morphing waverider aerodynamics, the Mars Science Laboratory (MSL) capsule, and the Space Shuttle. Three morphing waveriders were compared to the Space Shuttle, which resulted in reduced heating and peak deceleration. One morphing waverider was compared to the MSL capsule, which demonstrated a reduction in the peak stagnation heat flux, a reduction in the peak and average deceleration, and a reduction in the peak area-averaged heating

Optimisation of petaloid base dimensions and process operating conditions to minimize environmental stress cracking in injection stretch blow moulded PET bottles

Injection stretch blow moulded PET bottles are the most widely used container type for carbonated soft drinks. PET offers excellent clarity, good mechanical and barrier properties, and ease of processing. Typically, these bottles have a petaloid-shaped base, which gives good stability to the bottle and it is the most appropriate one for beverage storage. However, the base is prone to environmentally induced stress cracking and this a major concern to bottle manufacturers. The object of this study is to explain the occurrence of stress cracking, and to prevent it by optimising both the geometry of the petaloid base and the processing parameters during bottle moulding. A finite element model of the petaloid shape is developed in CATIA V5 R14, and used to predict the von Mises stress in the bottle base for different combinations of three key dimensions of the base: foot length, valley width, and clearance. The combination of dimensions giving the minimum stress is found by a statistical analysis approach using an optimisation and design of experiments software package ECHIP-7. A bottle mould was manufactured according to the optimum base geometry and PET bottles are produced by injection stretch blow moulding (ISBM). In order to minimise the stresses at the bottom of the bottle, the ISBM process parameters were reviewed and the effects of both the stretch rod movement and the temperature profile of the preform were studied by means of the process simulation software package (Blow View version 8.2). Simulated values of the wall thickness, stress, crystallinity, molecular orientation and biaxial ratio in the bottle base were obtained. The process parameters, which result in low stress and uniform material in the bottle base, are regarded as optimum operating conditions. In the evaluation process of the optimum bottle base, bottles with standard (current) and optimized (new) base were produced under the same process conditions via a two-stage ISBM machine. In order to compare both the bottles, environmental stress crack resistance, top load strength, burst pressure strength, thermal stability test as well as crystallinity studies ¬¬¬via modulated differential scanning calorimetry (MDSC) and morphology studies via environmental scanning electron microscopy (ESEM) and optical microscopy were conducted. In this study carried out, the new PET bottle with the optimised base significantly decreased the environmental stress cracking occurrence in the bottom of the bottle. It is found that the bottle with optimised base is stronger than the bottle with standard base against environmental stress cracking. The resistance time against environmental stress cracking are increased by about % 90 under the same operating process conditions used for standard (current) bottles; and by % 170 under the optimised process conditions where the preform re-heating temperature is set to 105 oC

Towards a Conceptual Design of an Intelligent Material Transport Based on Machine Learning and Axiomatic Design Theory

Reliable and efficient material transport is one of the basic requirements that affect productivity in sheet metal industry. This paper presents a methodology for conceptual design of intelligent material transport using mobile robot, based on axiomatic design theory, graph theory and artificial intelligence. Developed control algorithm was implemented and tested on the mobile robot system Khepera II within the laboratory model of manufacturing environment. Matlab© software package was used for manufacturing process simulation, implementation of search algorithms and neural network training. Experimental results clearly show that intelligent mobile robot can learn and predict optimal material transport flows thanks to the use of artificial neural networks. Achieved positioning error of mobile robot indicates that conceptual design approach can be used for material transport and handling tasks in intelligent manufacturing systems