Numerical quadrature for Gregory quads

We investigate quadrature rules in the context of quadrilateral Gregory patches, in short Gregory quads. We provide numerical and where possible symbolic quadrature rules for the space spanned by the twenty polynomial/rational functions associated with Gregory quads, as well as the derived spaces including derivatives, products, and products of derivatives of these functions. This opens up the possibility for a wider adoption of Gregory quads in numerical simulations

Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering

This paper concerns several important topics of the Symmetry journal, namely, computer-aided design, computational geometry, computer graphics, visualization, and pattern recognition. We also take advantage of the symmetric structure of the tensor-product surfaces, where the parametric variables u and v play a symmetric role in shape reconstruction. In this paper we address the general problem of global-support parametric surface approximation from clouds of data points for reverse engineering applications. Given a set of measured data points, the approximation is formulated as a nonlinear continuous least-squares optimization problem. Then, a recent metaheuristics called Cuckoo Search Algorithm (CSA) is applied to compute all relevant free variables of this minimization problem (namely, the data parameters and the surface poles). The method includes the iterative generation of new solutions by using the Lévy flights to promote the diversity of solutions and prevent stagnation. A critical advantage of this method is its simplicity: the CSA requires only two parameters, many fewer than any other metaheuristic approach, so the parameter tuning becomes a very easy task. The method is also simple to understand and easy to implement. Our approach has been applied to a benchmark of three illustrative sets of noisy data points corresponding to surfaces exhibiting several challenging features. Our experimental results show that the method performs very well even for the cases of noisy and unorganized data points. Therefore, the method can be directly used for real-world applications for reverse engineering without further pre/post-processing. Comparative work with the most classical mathematical techniques for this problem as well as a recent modification of the CSA called Improved CSA (ICSA) is also reported. Two nonparametric statistical tests show that our method outperforms the classical mathematical techniques and provides equivalent results to ICSA for all instances in our benchmark.This research work has received funding from the project PDE-GIR (Partial Differential Equations for Geometric modelling, Image processing, and shape Reconstruction) of the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No. 778035, the Spanish Ministry of Economy and Competitiveness (Computer Science National Program) under Grant #TIN2017-89275-R of the Agencia Estatal de Investigación and European Funds FEDER (AEI/FEDER, UE), and the project #JU12, jointly supported by public body SODERCAN of the Regional Government of Cantabria and European Funds FEDER (SODERCAN/FEDER UE). We also thank Toho University, Nihon University, and the Symmetry 2018, 10, 58 23 of 25 University of Cantabria for their support to conduct this research wor

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

Subdivision and manifold techniques for isogeometric design and analysis of surfaces

Design of surfaces and analysis of partial differential equations defined on them are of great importance in engineering applications, e.g., structural engineering, automotive and aerospace. This thesis focuses on isogeometric design and analysis of surfaces, which aims to integrate engineering design and analysis by using the same representation for both. The unresolved challenge is to develop a desirable surface representation that simultaneously satisfies certain favourable properties on meshes of arbitrary topology around the extraordinary vertices (EVs), i.e., vertices not shared by four quadrilaterals or three triangles. These properties include high continuity (geometric or parametric), optimal convergence in finite element analysis as well as simplicity in terms of implementation. To overcome the challenge, we further develop subdivision and manifold surface modelling techniques, and explore a possible scheme to combine the distinct appealing properties of the two. The unique advantages of the developed techniques have been confirmed with numerical experiments. Subdivision surfaces generate smooth surfaces from coarse control meshes of arbitrary topology by recursive refinement. Around EVs the optimal refinement weights are application-dependent. We first review subdivision-based finite elements. We then proceed to derive the optimal subdivision weights that minimise finite element errors and can be easily incorporated into existing implementations of subdivision schemes to achieve the same accuracy with much coarser meshes in engineering computations. To this end, the eigenstructure of the subdivision matrix is extensively used and a novel local shape decomposition approach is proposed to choose the optimal weights for each EV independently. Manifold-based basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. This thesis derives novel manifold-based basis functions with arbitrary prescribed smoothness using quasi-conformal maps, enabling us to model and analyse surfaces with sharp features, such as creases and corners. Their practical utility in finite element simulation of hinged or rigidly joined structures is demonstrated with Kirchhoff-Love thin shell examples. We also propose a particular manifold basis reproducing subdivision surfaces away from EVs, i.e., B-splines, providing a way to combine the appealing properties of subdivision (available in industrial software) for design and manifold basis (relatively new) for analysis.Cambridge International Scholarship Scheme (CISS) by Cambridge Trus

Extensions to OpenGL for CAGD.

Many computer graphic API’s, including OpenGL, emphasize modeling with rectangular patches, which are especially useful in Computer Aided Geomeric Design (CAGD). However, not all shapes are rectangular; some are triangular or more complex. This paper extends the OpenGL library to support the modeling of triangular patches, Coons patches, and Box-splines patches. Compared with the triangular patch created from degenerate rectangular Bezier patch with the existing functions provided by OpenGL, the triangular Bezier patches can be used in certain design situations and allow designers to achieve high-quality results that are less CPU intense and require less storage space. The addition of Coons patches and Box splines to the OpenGL library also give it more functionality. Both patch types give CAGD users more flexibility in designing surfaces. A library for all three patch types was developed as an addition to OpenGL

A geometrically exact isogeometric Kirchhoff plate: Feature‐preserving automatic meshing and C1 rational triangular Bézier spline discretizations

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/144603/1/nme5809.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/144603/2/nme5809_am.pd

Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity. © 2014 The Authors.EPSRC, Chinese Government (PhD studentship) and Cambridge Trust