► Hierarchical refinement of NURBS offers full analysis suitability, straightforward implementation and simple generalization to 3D. ► We first explore local hierarchical refinement for adaptive ...NURBS-based IGA. ► We then combine the B-spline version of the FCM and hierarchical refinement for a seamless design-through-analysis integration of 3D T-spline surface based models.
We explore hierarchical refinement of NURBS as a basis for adaptive isogeometric and immersed boundary analysis. We use the principle of B-spline subdivision to derive a local refinement procedure, which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions. We test hierarchical refinement of NURBS for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases. Using the B-spline version of the finite cell method, we illustrate the potential of immersed boundary methods as a seamless isogeometric design-through-analysis procedure for complex engineering parts defined by T-spline CAD surfaces, specifically a ship propeller and an automobile wheel. We show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features.
We develop a method for solving elliptic partial differential equations on surfaces described by CAD patches that may have gaps/overlaps. The method is based on hybridization using a ...three-dimensional mesh that covers the gap/overlap between patches. Thus, the hybrid variable is defined on a three-dimensional mesh, and we need to add appropriate normal stabilization to obtain an accurate solution, which we show can be done by adding a suitable term to the weak form. In practical applications, the hybrid mesh may be conveniently constructed using an octree to efficiently compute the necessary geometric information. We prove error estimates and present several numerical examples illustrating the application of the method to different problems, including a realistic CAD model.
•A robust method for elliptic PDE on multipatch CAD surfaces with gaps/overlaps.•Patches are weakly coupled via a hybrid variable on a 3D background grid.•The hybrid variable field is stabilized orthogonally to the interface curves.•Trimmed patches are facilitated using CutFEM stabilization techniques.
In this paper, the quasi-conforming method is introduced for the analysis of trimmed computer aided design (CAD) surfaces. The main benefit of the proposed method is that the boundary curves of ...elements are adopted for the numerical integration directly. In the quasi-conforming technique, the strains are approximated by using polynomials, and the weighted test functions are used to weaken the strain-displacement equations. The interpolation functions are introduced for strain integration. An appropriate choice of initial strain approximation and weighted test function ensures that inner-field functions are not required for strain integration, and this is used for the analysis of trimmed CAD surfaces. For example, the assumed stress quasi-conforming method is applied for the two-dimensional linear elastic problem. All the element edges are approximated by using quadratic Bézier curves for the conciseness, and this is easily incorporated into existing finite element codes and applied to Dirichlet boundary conditions. Numerical examples indicate the effectiveness and accuracy of the method.
•Quasi-conforming method is developed for the isogeometric analysis of trimmed CAD surfaces.•Boundary curves are used for the numerical integration of element.•The linear elastic problem is illustrated for the method.•Bézier curves are used for approximation of boundary curves.•The method can easily incorporate into existing finite element codes and apply Dirichlet boundary conditions.
A method to optimize triangular and quadrilateral meshes on parameterized surfaces is proposed. The optimization procedure relocates the nodes on the surface to improve the quality (smooth) and ...ensures that the elements are not inverted (untangle). We detail how to express any measure for planar elements in terms of the parametric coordinates of the nodes. The extended measures can be used to check the quality and validity of a surface mesh. Then, we detail how to optimize any Jacobian-based distortion measure to obtain smoothed and untangled meshes with the nodes on the surface. We prove that this method is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations. The examples show that the method can optimize meshes composed by a large number of inverted elements. Finally, the method can be extended to obtain high-order meshes with the nodes on the CAD surfaces.
We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is ...represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.
The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.
The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.
•Calderon's boundary equations and Huygens' principle to compute acoustic scattering.•Scattering surface represented by CAD with patched parameterization.•High-order accurate approximations to compute Calderon's operators.•Grid-independent computational complexity at the boundary.•Non-conforming geometries on Cartesian grids preserving accuracy and stability.
During the design of an object using a CAD (computer aided design) platform, the user can visualize the ongoing model at every moment. Visualization is based on a discrete representation of the model ...that coexists with the exact analytical representation of the object. Most CAD systems have this discrete representation available, and each of them applies its own construction methodology. This paper presents a new method to build a discrete (“triangulated” with quadrilaterals and triangles) model for CAD surfaces. It presents two major particularities: most elements are aligned with iso-parametric curves and the accuracy of the surface approximation is controlled. In addition, we present a new technique of surface mesh generation that is based on this discrete model. Several examples are presented to confirm the efficacy of this approach.
This paper presents a novel form design and CAD modelling approach to simultaneously support intuitive physical model development such as 3D sculpting, and CAD surface modelling for effective design ...communication, evaluation and collaboration. The concept of creation of 3D form profiles by simply projecting a set of 2D drawing on a physical model is utilised for generating 3D constraints and converting a single perspective image of the physical model captured by a web camera to a CAD surface model. This method has been implemented and tested in our prototype design desktop system.
Nasopharyngeal obstruction by adenoid enlargement is one of the main causes of mouth breathing. Cephalometric radiographs and rhinomanometric tests to evaluate nasal obstruction have been available ...for several decades. Various lines and areas have been interpreted by number of investigators to implicate the enlarged adenoid in a casual relationship with mouth breathing and the subsequent effect on vertical facial growth. The aim of this paper is to review lateral cephalometric tracing methods combined with newer Auto-cad surface area measurement program so that assessment of the nasopharyngeal free airway space can be done based on it, before more rigorous ear-nose-throat follow up is needed for the patient.
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