A new approach to free-form surface mesh control in a CAD environment

An adaptive process controlling the position of nodes on a surface mesh is presented. The control can depend on one (or more) criterion(ria) about element quality. The mesh is attached, through the concept of classification, to a geometric model issued by a computer aided design software. Thus, the surface domain is described by entities currently available in such systems, i.e. any free-form patches like Non-Uniform Rational B-Spline or Bézier patches can be used, even if they are restricted. Multi-connected surface domains can be treated using the same geometrical definition. The method described allows nodes to slide on a patch or jump from a patch onto another one. Such movements greatly improve the mesh quality with regard to a chosen criterion. Problems occurring with patch-by-patch meshing techniques when surfaces patches exhibit significant size differences are then overcome. The adaptation technique can be made independent of CAD data structures and meshing techniques, hence it constitutes the basis of a mesh management module.     

Bezier三角组合曲面的局域设计

 在三角组合曲面的设计中，需要根据曲面间的拼接条件，协调求解曲面的内部控制顶点．在分析Bezier三角组合曲面设计方法的基础上，提出了Bezier三角曲面GC 连续局域设计方法． 内部分割点采用九参数三次曲面设计方法估算，由内部分割点将三角曲面分割为3个子曲面片，通过构造曲面边界的跨界过渡切矢，推导得出了3个分割子曲面的内部控制顶点的代数表达式．应用该方法，可以简化三角组合曲面的设计过程，提高三角组合曲面的设计计算速度．     

Algebraic grid generation on trimmed parametric surface using non‐self‐overlapping planar Coons patch

Using a Coons patch mapping to generate a structured grid in the parametric region of a trimmed surface can avoid the singularity of elliptic PDE methods when only C¹ continuous boundary is given; the error of converting generic parametric C¹ boundary curves into a specified representation form is also avoided. However, overlap may happen on some portions of the algebraically generated grid when a linear or naïve cubic blending function is used in the mapping; this severely limits its usage in most of engineering and scientific applications where a grid system of non‐self‐overlapping is strictly required. To solve the problem, non‐trivial blending functions in a Coons patch mapping should be determined adaptively by the given boundary so that self‐overlapping can be averted. We address the adaptive determination problem by a functional optimization method. The governing equation of the optimization is derived by adding a virtual dimension in the parametric space of the given trimmed surface. Both one‐ and two‐parameter blending functions are studied. To resolve the difficulty of guessing good initial blending functions for the conjugate gradient method used, a progressive optimization algorithm is then proposed which has been shown to be very effective in a variety of practical examples. Also, an extension is added to the objective function to control the element shape. Finally, experiment results are shown to illustrate the usefulness and effectiveness of the presented method. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3D contact smoothing method using Gregory patches

In this work, a method is developed for smoothing three‐dimensional contact surfaces. The method can be applied to both regular and irregular meshes. The algorithm employs Gregory patches to interpolate finite element nodes and provide tangent plane continuity between adjacent patches. The resulting surface interpolation is used to calculate gaps and contact forces, in a variationally consistent way, such that contact forces due to normal and frictional contact vary smoothly as slave nodes transition from one patch to the next. This eliminates the ‘chatter’ which typically occurs in a standard contact algorithm when a slave node is situated near a master facet edge. The elimination of this chatter provides a significant improvement in convergence behaviour, which is illustrated by a number of numerical examples. Furthermore, smoothed surfaces also provide a more accurate representation of the actual surface, such that resulting stresses and forces can be more accurately computed with coarse meshes in many problems. This fact is also demonstrated by the numerical examples. Published in 2002 by John Wiley & Sons, Ltd.     

Surface characterization and parameter estimation for industrial vision system

In this paper we have presented some geometric techniques to characterize and parametrize surfaces of industrial parts in range images. The surfaces are characterized to one of plane, sphere, cylinder and cone, because they form the majority of object surfaces in man-made industrial parts. The problem has been studied for two different situations. In the first case,a priori knowledge about the surface shape is assumed. In such a situation the problem of surface characterization reduces to that of surface parameter estimation. The standard deviations of the estimated parameters give a measure of uncertainty of characterizing a surface patch to one of the four surface types. In the second case, noa priori information regarding the shape of a surface is available. This includes partially visible surfaces also. To deal with such a situation, a fuzzy classifier is designed using the uncertainty values. The fuzzy classifier classifies the unknown surface patch (including partially visible surfaces) to one of the four surface types. Experimental results with synthetic range images are presented to highlight the distinctive features of our technique.     

A higher‐order potential flow method for thick bodies,thin surfaces and wakes

A higher‐order panel method for analysing the three‐dimensional potential flow fields around bodies and wakes is presented. The geometric surfaces are represented by continuous curved patches, with no discretization into panels. These geometric patches hold singularity distributions that have C2 continuity, and which are solved by applying Dirichlet or Neumann boundary conditions at discrete collocation points. While higher‐order methods have previously been developed for thick bodies and Dirichlet boundary conditions, this potential flow method is capable of modelling continuous geometry and singularity surfaces over thin bodies and wakes. The continuous surface method has a number of advantages over conventional constant panel methods. Firstly, as curved geometries are represented exactly, changing the order of the solution does not modify the physical shape of the configuration under investigation. Furthermore, the continuous singularity distributions allow velocities to be evaluated accurately across the entire surface rather than just at collocation points. This means that pressure distributions can be calculated exactly without interpolation, and streamlines can be constructed very close to surfaces without problems of divergence. Finally, body and wake surfaces do not exhibit the strong modelling singularities that can present difficulties with wake relaxation. Copyright © 2007 John Wiley & Sons, Ltd.     

Parametric design and optimization of addendum surfaces for sheet metal forming process

In the sheet forming process, the addendum surfaces have an important influence on the formability and quality of the workpiece. Generally, the manual design of these surfaces (even with a CAD software) is very tedious and requires many trials-corrections. This study proposes an automatic design procedure based on a parametric method and a profile curve technique for the creation of addendum surfaces that involve the easiest process parameters to improve the part’s quality. All the addendum surfaces in form of NURBS are created without manual intervention. These geometric parameters are then optimized to obtain the best formability by using an optimization algorithm and the fast Inverse Approach for the sheet forming modelling. The performance and efficiency of two optimization algorithms (FSQP and RSM) are studied and compared. The parallel computation technique is used in the optimization procedure and the computation time is largely reduced. The present optimization procedure is applied to an academic example and two industrial workpieces. The numerical results show that the design is very fast and converges rapidly towards an optimal solution.     

GradH‐Correction: guaranteed sizing gradation in multi‐patch parametric surface meshing

In this paper a new method, called GradH‐Correction, for the generation of multi‐patch parametric surface meshes with controlled sizing gradation is presented. Such gradation is obtained performing a correction on the size values located on the vertices of the background mesh used to define the control space that governs the meshing process. In the presence of a multi‐patch surface, like shells of BREP solids, the proposed algorithm manages the whole composite surface simultaneously and as a unique entity. Sizing information can spread from a patch to its adjacent ones and the resulting size gradation is independent from the surface partitioning. Theoretical considerations lead to the assertion that, given a parameter λ, after performing a GradH‐Correction of level λ over the control space, the unit mesh constructed using the corrected control space is a mesh of gradation λ in the real space (target space). This means that the length ratio of any two adjacent edges of the mesh is bounded between 1/λ and λ. Numerical results show that meshes generated from corrected control spaces are of high quality and good gradation also when the background mesh has poor quality. However, due to mesh generator imprecision and theoretical limitations, guaranteed gradation is achieved only for the sizing specifications and not for the generated mesh. Copyright © 2004 John Wiley & Sons, Ltd.     

The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis

In this contribution, a mortar‐type method for the coupling of non‐conforming NURBS (Non‐Uniform Rational B‐spline) surface patches is proposed. The connection of non‐conforming patches with shared degrees of freedom requires mutual refinement, which propagates throughout the whole patch due to the tensor‐product structure of NURBS surfaces. Thus, methods to handle non‐conforming meshes are essential in NURBS‐based isogeometric analysis. The main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation. The deformations of the interface control points of adjacent patches are interrelated with a master‐slave relation. This relation is established numerically using the weak form of the equality of mutual deformations along the interface. With the help of this relation, the interface degrees of freedom of the slave patch can be condensated out of the system. A natural connection of the patches is attained without additional terms in the weak form. The proposed method is also applicable for nonlinear computations without further measures. Linear and geometrical nonlinear examples show the high accuracy and robustness of the new method. A comparison to reference results and to computations with the Lagrange multiplier method is given. Copyright © 2015 John Wiley & Sons, Ltd.     

An investigation into design of fair surfaces over irregular domains using data-dependent triangulation

Design of fair surfaces over irregular domains is a fundamental problem in computer-aided geometric design (CAGD), and has applications in engineering sciences (in aircraft, automobile, ship science etc.). In the design of fair surfaces over irregular domains defined over scattered data, it was widely accepted till recently that the classical Delaunay triangulation be used because of its global optimum property. However, in recent times it has been shown that for continuous piecewise linear surfaces, improvements in the quality of fit can be achieved if the triangulation pattern is made dependent upon some topological or geometric property of the data set or is simply data dependent. The fair surface is desired because it ensures smooth and continuous surface planar cuts, and these in turn ensure smooth and easy production of the surface in CAD/CAM, and favourable resistance properties. In this paper, we discuss a method for construction of C¹ piecewise polynomial parametric fair surfaces which interpolate prescribed ℜ³ scattered data using spaces of parametric splines defined on H³ triangulation. We show that our method is more specific to the cases when the projection on a 2-D plane may consist of triangles of zero area, numerically stable and robust, and computationally inexpensive and fast. Numerical examples dealing with surfaces approximated on plates, and on ships have been presented.     

Point-based approach to enhance the quality of geometric data for CAE applications

Commercial CAD surface modelling software is based almost exclusively on Bézier, B-spline and NURBS representations. These methods offer simple interactive shape modification and computationally efficient interrogations, but have some serious practical limitations. The root cause of all these problems is that the parametric polynomial methods are not geometry-based, and the trouble is compounded by the master geometry philosophy which treats the CAD model as if it were absolutely correct. An alternative purely geometric approach to surface modelling has been established, where the surface shape is characterised by a grid of points lying on the surface, but it is recognised that this approach would require major changes in existing CAE systems. This application paper explains the rationale to develop algorithms, which would plug into existing CAE software systems, so that the user could benefit from the point-based construction without disturbing the established mode of operation. The new point-based approach has achieved significant success in ensuring that form tools produce panels within a specified tolerance.     

On the Cauchy principal value of the surface integral in the boundary integral equation of 3D elasticity

A simple demonstration of the existence of the Cauchy principal value (CPV) of the strongly singular surface integral in the Somigliana Identity at a non-smooth boundary point is presented. First a regularization of the strongly singular integral by analytical integration of the singular term in the radial direction in pre-image planes of smooth surface patches is carried out. Then it is shown that the sum of the angular integrals of the characteristic of the tractions of the Kelvin fundamental solution is zero, a formula for the transformation of angles between the tangent plane of a suface patch and the pre-image plane at smooth mapping of the surface patch being derived for this purpose.     

Templatized refinement of triangle meshes using surface interpolation

Mesh refinement is an important process with regards to achieving good accuracy for computational simulation and analysis. Currently, there is a lack of a high‐fidelity refinement algorithm for the accurate modelling of geometry in the absence of a physical geometric model. This paper focuses on using a surface interpolation procedure based on a quartic triangular Bezier patch to approximate the underlying geometry of a mesh and to determine the locations of new subdivision vertices. A robust methodology is used for feature retention and accurate curve fitting at sharp edges and hard vertices. This extends the applicability of the surface fitting procedure to any arbitrary geometric configuration. The refinement is based on a new 1:9 subdivision scheme and its implementation is discussed in detail. Despite its high order subdivision footprint, computational efficiency is made possible by the effective use of lookup tables. Copyright © 2005 John Wiley & Sons, Ltd.     

T‐spline finite element method for the analysis of shell structures

A T‐spline surface is a nonuniform rational B‐spline (NURBS) surface with T‐junctions, and is defined by a control grid called T‐mesh. The T‐mesh is similar to a NURBS control mesh except that in a T‐mesh, a row or column of control points is allowed to terminate in the inner parametric space. This property of T‐splines makes local refinement possible. In the present study, shell formulation based on the T‐spline finite element method (FEM) is presented. Shell formulation based on NURBS or T‐splines has fundamental limitations because rotational DOFs, which are necessary in the shell formulation, cannot be defined on control points. In this study, the simple mapping scheme, in which every control point is mapped into one geometric point on the surface, is employed to eliminate the limitations. Using this mapping scheme, T‐spline FEM can be easily extended to the analysis of shells. The proposed shell formulation is verified through various benchmarking problems. This study is a part of the efforts by the authors for the integration of CAD–CAE processes. Copyright © 2009 John Wiley & Sons, Ltd.     

An extended advancing front technique for closed surfaces mesh generation

An extended advancing front technique (AFT) with shift operations and Riemann metric named as shifting‐AFT is presented for finite element mesh generation on 3D surfaces, especially 3D closed surfaces. Riemann metric is used to govern the size and shape of the triangles in the parametric space. The shift operators are employed to insert a floating space between real space and parametric space during the 2D parametric space mesh generation. In the previous work of closed surface mesh generation, the virtual boundaries are adopted when mapping the closed surfaces into 2D open parametric domains. However, it may cause the mesh quality‐worsening problem. In order to overcome this problem, the AFT kernel is combined with the shift operator in this paper. The shifting‐AFT can generate high‐quality meshes and guarantee convergence in both open and closed surfaces. For the shifting‐AFT, it is not necessary to introduce virtual boundaries while meshing a closed surface; hence, the boundary discretization procedure is largely simplified, and moreover, better‐shaped triangles will be generated because there are no additional interior constraints yielded by virtual boundaries. Comparing with direct methods, the shifting‐AFT avoids costly and unstable 3D geometrical computations in the real space. Some examples presented in this paper have demonstrated the advantages of shift‐AFT in 3D surface mesh generation, especially for the closed surfaces. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

Calculations on general point‐set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges. The absence of a mesh makes it nontrivial to decide if two neighboring points in the three‐dimensional embedding are nearby or rather far apart on the manifold. Furthermore, the topology of surfaces is generally not that of an open two‐dimensional set, ruling out global parametrizations. We propose a general and simple numerical method analogous to the mathematical theory of manifolds, in which the point‐set surface is described by a set of overlapping charts forming a complete atlas. We proceed in four steps: (1) partitioning of the node set into subregions of trivial topology; (2) automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods; (3) parametrization of the surface using smooth meshfree (here maximum‐entropy) approximants; and (4) gluing together the patch representations by means of a partition of unity. Each patch may be viewed as a meshfree macro‐element. We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchhoff–Love theory of thin‐shells. We analyze standard benchmark tests as well as point‐set surfaces of complex geometry and topology. Copyright © 2012 John Wiley & Sons, Ltd.     

Reissner–Mindlin shell implementation and energy conserving isogeometric multi‐patch coupling

A shear‐flexible isogeometric Reissner–Mindlin shell formulation using non‐uniform rational B‐splines basis functions is introduced, which is used for the demonstration of a coupling approach for multiple non‐conforming patches. The six degrees of freedom formulation uses the exact surface normal vectors and curvature. The shell formulation is implemented in an isogeometric analysis framework for computation of structures composed of multiple geometric entities. To enable local model refinement as well as non‐matching domains coupling, a conservative multi‐patch approach using Lagrange multipliers for structured non‐uniform rational B‐splines patches is presented. Here, an additional border frame mesh is used to couple geometries during structural analyses. This frame interface approach avoids the problem of excessive constraints when multiple patches are coupled at one point. First, the shell formulation is verified with several reference cases. Then the influence of the frame interface discretization and frame penalty stiffness on the smoothness of the results is investigated. The effects of the perturbed Lagrangian method in combination with the frame interface approach is shown. In addition, results of models with T‐joint interface connections and perpendicular stiffener patches are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

A non‐linear interface element for 3D mesoscale analysis of brick‐masonry structures

This paper presents a novel interface element for the geometric and material non‐linear analysis of unreinforced brick‐masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, whereas the mortar and brick–mortar interfaces are modelled by means of the 2D non‐linear interface element. This enables the representation of any 3D arrangement for brick‐masonry, accounting for the in‐plane stacking mode and the through‐thickness geometry, and importantly it allows the investigation of both the in‐plane and the out‐of‐plane responses of unreinforced masonry panels. A co‐rotational approach is employed for the interface element, which shifts the treatment of geometric non‐linearity to the level of discrete entities, and enables the consideration of material non‐linearity within a simplified local framework employing first‐order kinematics. In this respect, the internal interface forces are modelled by means of elasto‐plastic material laws based on work‐softening plasticity and employing multi‐surface plasticity concepts. Following the presentation of the interface element formulation details, several experimental–numerical comparisons are provided for the in‐plane and out‐of‐plane static behaviours of brick‐masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the non‐linear analysis of brick‐masonry structures under extreme loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.