Einstein equations in the null quasi-spherical gauge III: numerical algorithms

We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed

Almost-C1C^1 splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-$C^1$ are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are $C^1$ smooth almost everywhere, that is, at all vertices and across most edges, and in addition almost (i.e. approximately) $C^1$ smooth across all other edges. Thus, the splines form $H^2$-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-$C^1$ splines are described in an explicit B\'ezier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behavior

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms

A BEM based on the Bézier/Bernstein polynomial for acoustic waveguide modelization

42nd International Conference on Boundary Elements and other Mesh Reduction Methods, BEM/MRM 2019; ITeCons-University of CoimbraCoimbra; Portugal; 2 July 2019 through 4 July 2019; Code 155806. Publicado en WIT Transactions on Engineering Sciences, Vol 126This paper proposes a novel boundary element approach formulated on the Bézier–Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aided design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton–Bernstein algorithm. The applicability of the proposed method is demonstrated by solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.Ministerio de Economía y Competitividad BIA2016-75042-C2-1-

Smooth representation of thin shells and volume structures for isogeometric analysis

The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part. First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images. Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids. Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1 continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems

Multiscale modelling of woven and knitted fabric membranes

Light-weight fabric membranes have gained increasing popularity over the past years due to their tailorable structural and material performances. These tailorable properties include stretch forming and deep drawing formability that exhibits excellent stretchability and drapeability properties of textiles and textile composites. Since the inception of computerised numerical control for three-dimensional textile-manufacturing machines, technical textiles paved their way to numerous applications, certainly not limited to; aerospace, biomedical, civil engineering, defence, marine and medical industries. Digital interlooping and digital interlacing technology in additive manufacturing greatly advanced the manufacturing processes of textiles. In this work, we consider two branches of technical fabrics, namely plain-woven and weft-knitted. Multiscale modelling is the tool of choice for homogenising periodic structures and has been used extensively to model and analyse the mechanical behaviour of woven and knitted fabrics. But there is a plethora of literature discussing the demerits of such conventional multiscale modelling. These demerits include higher computational costs, rigid numerical models, ineffcient algorithmic computations and inability to incorporate geometric nonlinearities. We propose a data-driven nonlinear multiscale modelling technique to analyse the complex mechanical behaviour of plain-woven and weft-knitted fabrics with a neat extension to fabric material designing. We show how the integration of statistical learning techniques mitigates the weaknesses of conventional multiscale modelling. Moreover, we discuss the avenues that will open in many potential fields with regard to material modelling, structural engineering and textile industries. In the proposed data-driven nonlinear computational homogenisation technique, we effi ciently integrate the microscale and macroscale using Gaussian Process Regression (GPR) statistical learning technique. In the microscale, representative volume elements (RVEs) are modelled using nite deformable isogeometric spatial rods and deformation is homogenised using periodic boundary conditions. This nite deformable rod is profi cient in handling large deformations, rod-to-rod contacts, arbitrary cross-section de finitions and follower loads. Respecting the principle of separation of scales, we construct response databases by applying different homogenised strain states to the RVEs and recording the respective incremental volume-averaged energy values. We use GPR to learn a model using a 5-fold cross-validation technique by optimising the log marginal likelihood. In the macroscale, textiles are modelled as nonlinear orthotropic membranes for which the stresses and material constitutive relations are predicted by the trained GPR model. This coupling between GPR and membrane models is achieved through a systematic and seamless nite element integration using C++ and Python environments. A neat extension to material designing is also discussed with potentials to extend the work into other related fi elds.Cambridge trust and Trinity Hall scholarshi