Quasi-automatic 3D finite element model generation for individual single-rooted teeth and periodontal ligament

The paper demonstrates how to generate an individual 3D volume model of a human single-rooted tooth using an automatic workflow. It can be implemented into finite element simulation. In several computational steps, computed tomography data of patients are used to obtain the global coordinates of the tooth's surface. First, the large number of geometric data is processed with several self-developed algorithms for a significant reduction. The most important task is to keep geometrical information of the real tooth. The second main part includes the creation of the volume model for tooth and periodontal ligament (PDL). This is realized with a continuous free form surface of the tooth based on the remaining points. Generating such irregular objects for numerical use in biomechanical research normally requires enormous manual effort and time. The finite element mesh of the tooth, consisting of hexahedral elements, is composed of different materials: dentin, PDL and surrounding alveolar bone. It is capable of simulating tooth movement in a finite element analysis and may give valuable information for a clinical approach without the restrictions of tetrahedral elements. The mesh generator of FE software ANSYS executed the mesh process for hexahedral elements successfully.     

A strategy of automatic hexahedral mesh generation by using an improved whisker-weaving method with a surface mesh modification procedure

One of the demands for three dimensional (3D) finite element analyses is the development of an automatic hexahedral mesh generator. For this problem, several methods have been proposed by many researchers. However, reliable automatic hexahedral mesh generation has not been developed at present. In this paper, a new strategy of fully automatic hexahedral mesh generation is proposed. In this strategy, the prerequisite for generating a hexahedral mesh is a quadrilateral surface mesh. From the given surface mesh, combinatorial dual cycles (sheet loops for the whisker-weaving algorithm) are generated to produce a hexahedral mesh. Since generating a good quality hexahedral mesh does not depend only on the quality of quadrilaterals of the surface mesh but also on the quality of the sheet loops generated from it, a surface mesh modification method to remove self-intersections from sheet loops is developed. Next, an automatic hexahedral mesh generator by the improved whisker-weaving algorithm is developed in this paper. By creating elements and nodes on 3D real space during the weaving process, it becomes possible to generate a hexahedral mesh with fewer bad-quality elements. Several examples will be presented to show the validity of the proposed mesh generation strategy.     

混凝土三维钢筋埋置组合式有限单元模型及其网格自动生成

为克服三维钢筋混凝土有限单元模型中网格划分存在的缺陷，提出了埋置组合式有限单元模型。该模型在形成混凝土三维单元时可不考虑钢筋的方位，只要给出钢筋起点与终点的坐标，即可自动形成钢筋在组合单元中的信息，实际应用表明，该模型方便可行，单元划分效率较高。     

An algorithm for isochrome raster graphic display

Raster graphic display is used increasingly with microcomputers making use of readily available television monitors. An algorithm for obtaining the variation of a function on a screen by showing isochrome segments is presented. Small isoparametric bilinear elements are used as an adjustable intermediate size between the finite element mesh itself and the pixel. This approach permits a simple computer code that is adjustable to different computer hardware. A flow chart of the algorithm is presented and errors associated with the display are discussed. The results of two analyses are given. These examples were chosen for their differences both in the type of problem and in the numerical methods of solution used. First, the stresses in a gear tooth submitted to a concentrated load are presented. In this case, the algorithm is part of a post-processor following a finite element displacement analysis. A second example illustrates stream lines in a channel with a step. For the latter, the display shows the output data from a Gauss-Siedel resolution of a Laplace equation using a finite difference mesh.     

An algorithm for automatic 2D finite element mesh generation with line constraints

In this study, an algorithm is designed specifically for automatic finite element (FE) mesh generation on the transverse structure of hulls reinforced by stiffeners. Stiffeners attached to the transverse structure are considered as line constraints in the geometry boundary. For the FE mesh generation used in this study, the line constraints are treated as boundaries and by that means the geometry domain attached to the line constraints is decomposed into sub-domains, constrained only by the closed boundaries. Then, the mesh can be generated directly on those sub-domains by the traditional approach. The performance of the proposed algorithm is evaluated and the quality of the generated mesh meets expectations.     

Parallel multi-time step integration on a transputer system

An algorithm which allows different nodes of the finite element mesh to be integrated with different time steps is presented for second order finite element systems. The implementation of this algorithm on a system of transputer processors is discussed and a numerical example is used to evaluate the efficiency of the algorithm.     

Adaptive generation of hexahedral element mesh using an improved grid-based method

An improved grid-based algorithm for the adaptive generation of hexahedral finite element mesh is presented in this paper. It is named as the inside-out grid-based method and involves the following four steps. The first step is the generation of an initial grid structure which envelopes the analyzed solid model completely. And the elements size and density maps are constructed based on the surface curvature and local thickness of the solid model. Secondly, the core mesh is generated through removing all the undesired elements using even and odd parity rules. The third step is to magnify the core mesh in an inside-out manner through a surface node projection process using the closest position approach. To match the mesh to the characteristic boundary of the solid model, a minimal Scaled Jacobian criterion is employed. Finally, in order to handle the degenerated elements and improve the quality of the resulting mesh, two comprehensive techniques are employed: the insertion technique and collapsing technique. The present method was applied in the mesh construction of different engineering problems. Scaled Jacobian and Skew metrics are used to evaluate the hexahedral element mesh quality. The application results show that all-hexahedral element meshes which are well-shaped and capture all the geometric features of the original solid models can be generated using the inside-out grid-based method presented in this paper.     

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

有限元网格图拓扑分析

依据图论的方法对有限元网格图进行了拓朴分析，讨论了单元节点间的相关性，提出了构造单元网格节点拓扑阵和组集整体网格节点拓扑阵的方法。这是一种新的有限元网格图自动生成方法，简洁明快，具有良好的通用性。     

An open source program to generate zero-thickness cohesive interface elements

An open source program to generate zero-thickness cohesive interface elements in existing finite element discretizations is presented. This contribution fills the gap in the literature that, to the best of the author’s knowledge, there is no such program exists. The program is useful in numerical modeling of material/structure failure using cohesive interface elements. The program is able to generate one/two dimensional, linear/quadratic cohesive elements (i) at all inter-element boundaries, (ii) at material interfaces and (iii) at grain boundaries in polycrystalline materials. Algorithms and utilization of the program is discussed. Several two dimensional and three dimensional fracture mechanics problems are given including debonding process of material interfaces, multiple delamination of composite structures, crack propagation in polycrystalline structures.     

A three-dimensional self-adaptive cohesive zone model for interfacial delamination

Discrete crack models with cohesive binding forces in the fracture process zone have been widely used to address failure in quasi-brittle materials and interfaces. However, the numerical concerns and limitations stemming from the application of interface cohesive zone models in a quasi-static finite element framework increase considerably as the relative size of the process zone decreases. An excessively fine mesh is required in the process zone to accurately resolve the distribution of tractions in a relatively small moving zone. With a moderate mesh size, inefficient path-following techniques have to be employed to trace the local discretization-induced snap-backs. In order to increase the applicability of cohesive zone models by reducing their numerical deficiencies, a self-adaptive finite element framework is proposed, based on a hierarchical enrichment of the standard elements. With this approach, the planar mixed-mode crack growth in a general three-dimensional continuum, discretized by a coarse mesh, can be modeled while the set of equations of the non-linear system is solved by a standard Newton–Raphson iterative procedure. This hierarchical scheme was found to be most effective in reducing the oscillatory behavior of the global response.     

Triangular thick plate elements based on a hybrid-Trefftz approach

The paper presents a family of triangular, thick plate elements derived using the hybrid-Trefftz approach. Exact solutions of the governing thick plate equations are used as interpolations for the internal element displacements. An immediate benefit of this approach is that the locking problem is avoided a priori. Independent interpolations are used to describe the displacement and rotations on the element boundaries. The element formulation is based on a modified hybrid-stress principle, leading to a standard stiffness formulation. This enables the elements to be readily implemented into existing finite element schemes. A number of examples are considered to demonstrate the accuracy achieved by the elements.     

Numerical comparison between two cost functions in contact shape optimization

A finite element approach to shape optimization in a 2D frictionless contact problem for two different cost functions is presented in this work. The goal is to find an appropriate shape for the contact boundary, performing an almost constant contact-stress distribution. The whole formulation, including the mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by a direct modification of the geometry throughB-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data. Augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization) are used to obtain numerical results.     

Automatic mesh generation and transformation for topology optimization methods

This paper presents automatic tools aimed at the generation and adaptation of unstructured tetrahedral meshes in the context of composite or heterogeneous geometry. These tools are primarily intended for applications in the domain of topology optimization methods but the approach introduced presents great potential in a wider context. Indeed, various fields of application can be foreseen for which meshing heterogeneous geometry is required, such as finite element simulations (in the case of heterogeneous materials and assemblies, for example), animation and visualization (medical imaging, for example). Using B-Rep concepts as well as specific adaptations of advancing front mesh generation algorithms, the mesh generation approach presented guarantees, in a simple and natural way, mesh continuity and conformity across interior boundaries when trying to mesh a composite domain. When applied in the context of topology optimization methods, this approach guarantees that design and non-design sub-domains are meshed so that finite elements are tagged as design and non-design elements and so that continuity and conformity are guaranteed at the interface between design and non-design sub-domains. The paper also presents how mesh transformation and mesh smoothing tools can be successfully used when trying to derive a functional shape from raw topology optimization results.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.     

Optimality criterion approach using tapered finite elements with nodal averaging technique for two-dimensional problems

Use of tapered finite elements with three design variables per element and nodal averaging technique in the optimality criterion approach is studied in this paper. Minimum volume design of a uniformly heated square plate with a single temperature constraint is obtained using the proposed method. The present results with a lower order finite element mesh compare very well with those obtained with optimality criterion approach using constant thickness elements and with mathematical programming techniques using a higher order finite element mesh.     

An in-core grid index for transferring finite element data across dissimilar meshes

The simulation of a manufacturing process chain with the finite element method requires the selection of an appropriate finite element solver, element type and mesh density for each process of the chain. When the simulation results of one step are needed in a subsequent one, they have to be interpolated and transferred to another model. This paper presents an in-core grid index that can be created on a mesh represented by a list of nodes/elements. Finite element data can thus be transferred across different models in a process chain by mapping nodes or elements in indexed meshes. For each nodal or integration point of the target mesh, the index on the source mesh is searched for a specific node or element satisfying certain conditions, based on the mapping method. The underlying space of an indexed mesh is decomposed into a grid of variable-sized cells. The index allows local searches to be performed in a small subset of the cells, instead of linear searches in the entire mesh which are computationally expensive. This work focuses on the implementation and computational efficiency of indexing, searching and mapping. An experimental evaluation on medium-sized meshes suggests that the combination of index creation and mapping using the index is much faster than mapping through sequential searches.