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The purpose of this paper is to apply rectangular Bézier surface patches directly into the mathematical formula used to solve boundary value problems modeled by Laplace's equation. The mathematical formula, called the parametric integral equation systems (PIES), will be obtained through the analytical modification of the conventional boundary integral equations (BIE), with the boundary mathematically described by rectangular Bézier patches.

The paper presents the methodology of the analytic connection of the rectangular patches with BIE. This methodology is a generalization of the one previously used for 2D problems.

In PIES the paper separates the necessity of performing simultaneous approximation of both boundary shape and the boundary functions, as the boundary geometry has been included in its mathematical formalism. The separation of the boundary geometry from the boundary functions enables to achieve an independent and more effective improvement of the accuracy of both approximations. Boundary functions are approximated by the Chebyshev series, whereas the boundary is approximated by Bézier patches.

The originality of the proposed approach lies in its ability to automatic adapt the PIES formula for modified shape of the boundary modeled by the Bézier patches. This modification does not require any dividing the patch into elements and creates the possibility for effective declaration of boundary geometry in continuous way directly in PIES.

To focus on grid generation which is an essential part of any analytical tool for effective discretization.

This paper explores the application of the possibility of unstructured triangular grid generation that deals with derivationally continuous, smooth, and fair triangular elements using piecewise polynomial parametric surfaces which interpolate prescribed R³ scattered data using spaces of parametric splines defined on R² triangulations in the case of surfaces in engineering sciences. The method is based upon minimizing a physics‐based certain natural energy expression over the parametric surface. The geometry is defined as a set of stitched triangles prior to the grid generation. As for derivational continuities between the two triangular patches C⁰ and C¹ continuity or both, as per the requirements, has been imposed. With the addition of a penalty term, C² (approximate) continuity can also be achieved. Since, in this work physics‐based approach has been used, the grid is analyzed using intersection curves with three‐dimensional planes, and intrinsic geometric properties (i.e. directional derivatives), for derivational continuity and smoothness.

The triangular grid generation that deals with derivationally continuous, smooth, and fair triangular elements has been implemented in this paper for surfaces in engineering sciences.

This paper deals with the important problem of grid generation which is an essential part of any analytical tool for effective discretization. And, the examples to demonstrate the theoretical model of this paper have been chosen from different branches of engineering sciences. Hence, the results of this paper are of practical importance for grid generation in engineering sciences.

The paper is theoretical with worked examples chosen from engineering sciences.

This study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C¹ basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.

The weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C⁰ non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C¹) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.

It is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.

Fourth-order PFM is implemented using C¹ non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

Gives a bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore aims to give the reader an encyclopaedic view on the subject. The bibliography at the end of the paper contains 2,025 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1992‐1995.

Given an intersection-free mesh surface S, the paper introduces a method to thicken S into a solid H located at one side of S. By such a surface-to-solid conversion operation, industrial users are able to fabricate a designed (or reconstructed) surface by rapid prototyping.

The paper first investigates an implicit representation of the thickened solid H according to an extension of signed distance function. After that, a partial surface reconstruction algorithm is proposed to generate the boundary surface of H, which retains the given surface S on the resultant surface.

Experimental tests show that the thickening results generated by the method give nearly uniform thickness and meanwhile do not present shape approximation error at the region of input surface S. These two good properties are important to the industrial applications of solid fabrication.

The input polygonal model is assumed to be intersection-free, where models containing self-intersection will lead to invalid thickening results.

A novel robust operation is to convert a freeform open surface into a solid by introducing no shape approximation error. A new implicit function gives a compact mathematical representation, which can easily handle the topological change on the thickened solids. A new polygonization algorithm generates faces for the boundary of thickened solid meanwhile retaining faces on the input open mesh.

As computer simulation increasingly supports engineering design and manufacture, the requirement for a computer software environment providing an integration platform for computational engineering software increases. The potential benefits to industry are considerable. As a first step in the long‐term development of such a system, a computer software environment has been developed for pre‐ and post‐processing for unstructured grid‐based computational simulation. Arbitrary computer application software can be integrated into the environment to provide a multi‐disciplinary engineering analysis capability within one unified computational framework. Recognising the computational demands of many application areas, the environment includes a set of parallel tools to help the user maximise the potential of high performance computers and networks. The paper will present details of the environment and include an example of, and discussion about, the integration of application software.

Thin-walled structures inevitably always have manufacturing deviations, which affects the assembly quality of mechanical products. The assembly quality directly determines the performances, reliability and service life of the products. To achieve the automatic assembly of large-scale thin-walled structures, the sizing force of the structures with deviations should be calculated, and its assembling ability should be studied before assembly process. The purpose of this study is to establish a precise model to describe the deviations of structures and to study the variation propagation during assembly process.

Curved thin-walled structures are modeled by using the shell element via the absolute nodal coordinate formulation. Two typical deviation modes of the structure are defined. The generalized elastic force of shell elements with anisotropic materials is deduced based on a continuum mechanics approach to account for the geometric non-linearity. The quasi-static method is introduced to describe the assembly process. The effects of the deviation forms, geometrical parameters of the thin-walled structures and material properties on assembly quality are investigated numerically.

The geometric non-linearity of structure and anisotropy of materials strongly affect the variation propagation and the assembly quality. The transformation and accumulation effects of the deviations are apparent in the multiple assembly process. The constraints on the structures during assembly can reduce assembly deviation.

The plate element via the absolute nodal coordinate formulation is first introduced to the variation propagation analysis. Two typical shape deviation modes are defined. The elastic force of structures with anisotropic materials is deduced. The variation propagation during the assembly of structures with various geometrical and material parameters is investigated.

In this paper, we are interested in the forming of composite part by deep‐drawing and laying‐up processes. We present a new finite element model for the simulation of these processes. The augmented Lagrangian approach is adopted to treat the frictional contact between the composite fabric and the tools. It is based on a new way of writing the Coulomb’s friction law. The numerical simulation is carried out with Abaqus/Explicit software and some numerical results are given to validate the proposed numerical method.

Gives a bibliographical review of the finite element meshing and remeshing from the theoretical as well as practical points of view. Topics such as adaptive techniques for meshing and remeshing, parallel processing in the finite element modelling, etc. are also included. The bibliography at the end of this paper contains 1,727 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1990 and 2001.

Arbitrary constraints might be included into the problem domain in many engineering applications, which represent specific features such as multi-domain interfaces, cracks with small yield stresses, stiffeners attached on the plate for reinforcement and so on. To imprint these constraints into the final mesh, additional techniques need to be developed to treat these constraints properly.

This paper proposes an automatic approach to generate quadrilateral meshes for the geometric models with complex feature constraints. Firstly, the region is decomposed into sub-regions by the constraints, and then the quadrilateral mesh is generated in each sub-region that satisfies the constraints. A method that deals with constraint lines and points is presented. A distribution function is proposed to represent the distribution of mesh size over the region by using the Laplace equation. The density lines and points can be specified inside the region and reasonable mesh size distribution can be obtained by solving the Laplace equation.

An automatic method to define sub-regions is presented, and the user interaction can be avoided. An algorithm for constructing loops from constraint lines is proposed, which can deal with the randomly distributed constraint lines in a general way. A method is developed to deal with constraint points and quality elements can be generated around constraint points. A function defining the distribution of mesh size is put forward. The examples of constrained quadrilateral mesh generation in actual engineering analysis are presented to show the performance of the approach.

An automatic approach to constrained quadrilateral mesh generation is presented in this paper. It can generate required quality meshes for special applications with complex internal feature constraints.