Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures

A novel surface interrogation technique is proposed to compute the intersection of curves with spline surfaces in isogeometric analysis. The intersection points are determined in one-shot without resorting to a Newton-Raphson iteration or successive refinement. Surface-curve intersection is required in a wide range of applications, including contact, immersed boundary methods and lattice-skin structures, and requires usually the solution of a system of nonlinear equations. It is assumed that the surface is given in form of a spline, such as a NURBS, T-spline or Catmull-Clark subdivision surface, and is convertible into a collection of B\'ezier patches. First, a hierarchical bounding volume tree is used to efficiently identify the B\'ezier patches with a convex-hull intersecting the convex-hull of a given curve segment. For ease of implementation convex-hulls are approximated with k-dops (discrete orientation polytopes). Subsequently, the intersections of the identified B\'ezier patches with the curve segment are determined with a matrix-based implicit representation leading to the computation of a sequence of small singular value decompositions (SVDs). As an application of the developed interrogation technique the isogeometric design and analysis of lattice-skin structures is investigated. The skin is a spline surface that is usually created in a computer-aided design (CAD) system and the periodic lattice to be fitted consists of unit cells, each containing a small number of struts. The lattice-skin structure is generated by projecting selected lattice nodes onto the surface after determining the intersection of unit cell edges with the surface. For mechanical analysis, the skin is modelled as a Kirchhoff-Love thin-shell and the lattice as a pin-jointed truss. The two types of structures are coupled with a standard Lagrange multiplier approach

Infill topology and shape optimisation of lattice-skin structures

Lattice-skin structures composed of a thin-shell skin and a lattice infill are widespread in nature and large-scale engineering due to their efficiency and exceptional mechanical properties. Recent advances in additive manufacturing, or 3D printing, make it possible to create lattice-skin structures of almost any size with arbitrary shape and geometric complexity. We propose a novel gradient-based approach to optimising both the shape and infill of lattice-skin structures to improve their efficiency further. The respective gradients are computed by fully considering the lattice-skin coupling while the lattice topology and shape optimisation problems are solved in a sequential manner. The shell is modelled as a Kirchhoff-Love shell and analysed using isogeometric subdivision surfaces, whereas the lattice is modelled as a pin-jointed truss. The lattice consists of many cells, possibly of different sizes, with each containing a small number of struts. We propose a penalisation approach akin to the SIMP (solid isotropic material with penalisation) method for topology optimisation of the lattice. Furthermore, a corresponding sensitivity filter and a lattice extraction technique are introduced to ensure the stability of the optimisation process and to eliminate scattered struts of small cross-sectional areas. The developed topology optimisation technique is suitable for non-periodic, non-uniform lattices. For shape optimisation of both the shell and the lattice, the geometry of the lattice-skin structure is parameterised using the free-form deformation technique. The topology and shape optimisation problems are solved in an iterative, sequential manner. The effectiveness of the proposed approach and the influence of different algorithmic parameters are demonstrated with several numerical examples.Comment: 20 pages, 17 figure

Isogeometric Design, Analysis and Optimisation of Lattice-Skin Structures

The advancements in additive manufacturing techniques enable novel designs using lattice structures in mechanical parts, lightweight materials, biomaterials and so forth. Lattice-skin structures are a class of structures that couple thin-shells with lattices, which potentially combine the advantages of the thin-shell and the lattice structure. A new and systematic isogeometric analysis approach that integrates the geometric design, structural analysis and optimisation of lattice-skin structures is proposed in the dissertation. In the geometric design of lattice-skin structures, a novel shape interrogation scheme for splines, specifically subdivision surfaces, is proposed, which is able to compute the line/surface intersection efficiently and robustly without resorting to successive refinements or iterations as in Newton-Raphson method. The line/surface intersection algorithm involves two steps: intersection detection and intersection computation. In the intersection detection process, a bounding volume tree of k-dops (discrete oriented polytopes) for the subdivision surface is first created in order to accelerate the intersection detection between the line and the surface. The spline patches which are detected to be possibly intersected by the line are converted to Bézier representations. For the intersection computation, a matrix-based algorithm is applied, which converts the nonlinear intersection computation into solving a sequence of linear algebra problems using the singular value decomposition (SVD). Finally, the lattice-skin geometry is generated by projecting selected lattice nodes to the nearest intersection points intersected by the lattice edges. The Stanford bunny example demonstrates the efficiency and accuracy of the developed algorithm. The structural analysis of lattice-skin structures follows the isogeometric approach, in which the thin-shell is discretised with spline basis functions and the lattice structure is modelled with pin-jointed truss elements. In order to consider the lattice-skin coupling, a Lagrange multiplier approach is implemented to enforce the displacement compatibility between the coupled lattice nodes and the thin-shell. More importantly, the parametric coordinates of the coupled lattice nodes on the thin-shell surface are obtained directly from the lattice-skin geometry generation, which integrates the design and analysis process of lattice-skin structures. A sandwich plate example is analysed to verify the implementation and the accuracy of the lattice-skin coupling computation. In addition, a SIMP-like lattice topology optimisation method is proposed. The topology optimisation results of lattice structures are analysed and compared with several examples adapted from the benchmark examples commonly used in continuum topology optimisation. The SIMP-like lattice topology optimisation proposed is further applied to optimise the lattice in lattice-skin structures. The lattice-skin topology optimisation is fully integrated with the lattice-skin geometry design since the sensitivity analysis in the proposed method is based on lattice unit cells which are inherited from the geometry design stage. Finally, shape optimisation of lattice-skin structures using the free-form deformation (FFD) technique is studied. The corresponding shape sensitivity of lattice-skin structures is derived. The geometry update of the lattice-skin structure is determined by the deformation of the FFD control volume, and in this process the coupling between lattice nodes and the thin-shell is guaranteed by keeping the parametric coordinates of coupled lattice nodes which are obtained in the lattice-skin geometry design stage. A pentagon roof example is used to explore the combination of lattice topology optimisation and shape optimisation of lattice-skin structures

Manifold-based isogeometric analysis basis functions with prescribed sharp features

We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed $C^0$ continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in $\mathbb R^3$ is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in $\mathbb R^2$, so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always $C^0$ continuous across tagged edges. The derived non-rational manifold-based basis functions are particularly well suited for isogeometric analysis of Kirchhoff-Love thin shells with kinks

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

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

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

An efficient active B-spline/nurbs model for virtual sculpting

This thesis presents an Efficient Active B-Spline/Nurbs Model for Virtual Sculpting. In spite of the on-going rapid development of computer graphics and computer-aided design tools, 3D graphics designers still rely on non-intuitive modelling procedures for the creation and manipulation of freeform virtual content. The ’Virtual Sculpting' paradigm is a well-established mechanism for shielding designers from the complex mathematics that underpin freeform shape design. The premise is to emulate familiar elements of traditional clay sculpting within the virtual design environment. Purely geometric techniques can mimic some physical properties. More exact energy-based approaches struggle to do so at interactive rates. This thesis establishes a unified approach for the representation of physically aware, energy-based, deformable models, across the domains of Computer Graphics, Computer-Aided Design and Computer Vision, and formalises the theoretical relationships between them. A novel reformulation of the computer vision approach of Active Contour Models (ACMs) is proposed for the domain of Virtual Sculpting. The proposed ACM-based model offers novel interaction behaviours and captures a compromise between purely geometric and more exact energy-based approaches, facilitating physically plausible results at interactive rates. Predefined shape primitives provide features of interest, acting like sculpting tools such that the overall deformation of an Active Surface Model is analogous to traditional clay modelling. The thesis develops a custom-approach to provide full support for B-Splines, the de facto standard industry representation of freeform surfaces, which have not previously benefited from the seamless embodiment of a true Virtual Sculpting metaphor. A novel generalised computationally efficient mathematical framework for the energy minimisation of an Active B-Spline Surface is established. The resulting algorithm is shown to significantly reduce computation times and has broader applications across the domains of Computer-Aided Design, Computer Graphics, and Computer Vision. A prototype ’Virtual Sculpting’ environment encapsulating each of the outlined approaches is presented that demonstrates their effectiveness towards addressing the long-standing need for a computationally efficient and intuitive solution to the problem of interactive computer-based freeform shape design