Universal meshes: A method for triangulating planar curved domains immersed in nonconforming meshes

We introduce a new method to triangulate planar, curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary. In the process, no new vertices are introduced, and connectivities of triangles are left unaltered. The method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection. To guarantee its robustness, we require that the domain be C²‐regular, the background mesh be sufficiently refined near the boundary, and that specific angles in triangles near the boundary be strictly acute. The method can render both straight‐edged and curvilinear triangulations for the immersed domain. The latter includes curved triangles that conform exactly to the immersed boundary, and ones constructed with isoparametric mappings to interpolate the boundary at select points. High‐order finite elements constructed over these curved triangles achieve optimal accuracy, which has customarily proven difficult in numerical schemes that adopt nonconforming meshes. Aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes, the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains. With no conformity requirements, the same background mesh can be adopted to triangulate a large family of domains immersed in it, including ones realized over several updates during the coarse of simulating problems with moving boundaries. We term such a background mesh as a universal mesh for the family of domains it can be used to triangulate. Universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes. Furthermore, because the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high‐order accuracy. We present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with Stokesian fluids. Copyright © 2014 John Wiley & Sons, Ltd.     

Universal meshes for smooth surfaces with no boundary in three dimensions

We introduce a method to mesh the boundary Γ of a smooth, open domain in immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh 1 for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved immersed moving boundary for hydrodynamic force calculation in lattice Boltzmann method

Particles suspension is considerably prevalent in petroleum industry and chemical engineering. The efficient and accurate simulation of such a process is always a challenge for both the traditional computational fluid dynamics and lattice Boltzmann method. Immersed moving boundary (IMB) method is promising to resolve this issue by introducing a particle-fluid interaction term in the standard lattice Boltzmann equation, which allows for the smooth hydrodynamic force calculation even for a large grid size relative to the solid particle. Although the IMB method was proved good for stationary particles, the deviation of hydrodynamic force on moving particles exists. In this work, we reveal the physical origin of this problem first and figure out that the internal fluid effect on the hydrodynamic force calculation is not counted in the previous IMB. An improved immersed moving boundary method is therefore proposed by considering the internal fluid correction, which is easy to implement with the little extra computation cost. A 2D single elliptical particle and a 3D sphere sedimentation in Newtonian fluid is simulated directly for the validation of the corrected model by excellent agreements with the standard data.     

Three‐dimensional immersed finite element methods for electric field simulation in composite materials

This paper presents two immersed finite element (IFE) methods for solving the elliptic interface problem arising from electric field simulation in composite materials. The meshes used in these IFE methods can be independent of the interface geometry and position; therefore, if desired, a structured mesh such as a Cartesian mesh can be used in an IFE method to simulate 3‐D electric field in a domain with non‐trivial interfaces separating different materials. Numerical examples are provided to demonstrate that the accuracies of these IFE methods are comparable to the standard linear finite element method with unstructured body‐fit mesh. Copyright © 2005 John Wiley & Sons, Ltd.     

An improved moving‐least‐squares reconstruction for immersed boundary method

The current work presents an improved immersed boundary method based on the ideas proposed by Vanella and Balaras (M. Vanella, E. Balaras, A moving‐least‐squares reconstruction for embedded‐boundary formulations, J. Comput. Phys. 228 (2009) 6617–6628). In the method, an improved moving‐least‐squares approximation is employed to build the transfer functions between the Lagrangian points and discrete Eulerian grid points. The main advantage of the improved method is that there is no need to obtain the inverse matrix, which effectively eliminates numerical instabilities caused by matrix inversion and reduces the computational cost significantly. Several different flow problems (Taylor‐Green decaying vortices, flows past a stationary circular cylinder and a sphere, and the sedimentation of a free‐falling sphere in viscous fluid) are simulated to validate the accuracy and efficiency of the method proposed in the present paper. The simulation results show good agreement with previous numerical and experimental results, indicating that the improved immersed boundary method is efficient and reliable in dealing with the fluid–solid interaction problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A stable interface-enriched formulation for immersed domains with strong enforcement of essential boundary conditions

Generating matching meshes for finite element analysis is not always a convenient choice, for instance, in cases where the location of the boundary is not known a priori or when the boundary has a complex shape. In such cases, enriched finite element methods can be used to describe the geometric features independently from the mesh. The Discontinuity-Enriched Finite Element Method (DE-FEM) was recently proposed for solving problems with both weak and strong discontinuities within the computational domain. In this paper, we extend DE-FEM to treat fictitious domain problems, where the mesh-independent boundaries might either describe a discontinuity within the object, or the boundary of the object itself. These boundaries might be given by an explicit expression or an implicit level set. We demonstrate the main assets of DE-FEM as an immersed method by means of a number of numerical examples; we show that the method is not only stable and optimally convergent but, most importantly, that essential boundary conditions can be prescribed strongly.     

An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation

A new hybrid algorithm for automatically generating either an all-quadrilateral or an all-triangular element mesh within an arbitrarily shaped domain is described. The input consists of one or more closed loops of straight-line segments that bound the domain. Internal mesh density is inferred from the boundary density using a recursive spatial decomposition (quadtree) procedure. All-triangular element meshes are generated using a boundary contraction procedure. All-quadrilateral element meshes are generated by modifying the boundary contraction procedure to produce a mixed element mesh at half the density of the final mesh and then applying a polygon-splitting procedure. The final meshes exhibit good transitioning properties and are compatible with the given boundary segments which are not altered. The algorithm can support discrete crack growth simulation wherein each step of crack growth results in an arbitrarily shaped region of elements deleted about each crack tip. The algorithm is described and examples of the generated meshes are provided for a representative selection of cracked and uncracked structures.     

An efficient Wave Based Method for solving Helmholtz problems in three-dimensional bounded domains

This paper discusses the use of the Wave Based Method for the analysis of time-harmonic three-dimensional (3D) interior acoustic problems. Conventional element-based prediction methods, such as the Finite Element Method, are most commonly used for these types of problems, but they are restricted to low-frequency applications. The Wave Based Method is an alternative deterministic technique which is based on the indirect Trefftz approach. Up to now, this method's very high computational efficiency has been illustrated mainly for two-dimensional (2D) problem settings, allowing the analysis of problems at higher frequencies. The numerical validation examples presented in this work shows that the enhanced computational efficiency of the Wave Based Method in comparison with conventional element-based methods is kept when the method is extended to 3D case with and without the presence of material damping.     

A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes

In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element-discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations. Copyright © 2012 John Wiley & Sons, Ltd.     

An automatic method for generating carbon nanostructure atomistic models using hexagonal meshes with properly distributed defects

Atomistic models, which are crucial for performing molecular dynamics simulations of carbon nanostructures, consist of virtual hexagonal meshes with defects properly distributed in the intersectional areas. Currently, atomistic models are created mostly by hand, which is a notably tedious and time‐consuming process. In this paper, we develop a method that produces atomistic models automatically. Because a hexagonal mesh and triangulation represent dual graphs, our work focuses on the creation of proper triangulation. The edge lengths of the triangulation should be compatible with the lengths of the C–C bonds, and vertices with valences other than 6 (due to the defects in the hexagonal mesh) should be properly arranged around the boundaries of the different components of a carbon nanostructure. Two techniques play important roles in our method: (1) sphere packing is used to place the nodes for triangulation that produces nearly constant edge lengths of the triangles and (2) the movement and editing of defects is used to control the number and positions of the defects. We subsequently develop a computer program based on this method that can create models much easier and faster than the current handwork method, thereby reducing the operation time significantly. Copyright © 2016 John Wiley & Sons, Ltd.     

Automatic metric 3D surface mesh generation using subdivision surface geometrical model. Part 2: Mesh generation algorithm and examples

In this paper, a new metric advancing front surface mesh generation scheme is suggested. This new surface mesh generator is based on a new geometrical model employing the interpolating subdivision surface concept. The target surfaces to be meshed are represented implicitly by interpolating subdivision surfaces which allow the presence of various sharp and discontinuous features in the underlying geometrical model. While the main generation steps of the new generator are based on a robust metric surface triangulation kernel developed previously, a number of specially designed algorithms are developed in order to combine the existing metric advancing front algorithm with the new geometrical model. As a result, the application areas of the new mesh generator are largely extended and can be used to handle problems involving extensive changes in domain geometry. Numerical experience indicates that, by using the proposed mesh generation scheme, high quality surface meshes with rapid varying element size and anisotropic characteristics can be generated in a short time by using a low‐end PC. Finally, by using the pseudo‐curvature element‐size controlling metric to impose the curvature element‐size requirement in an implicit manner, the new mesh generation procedure can also generate finite element meshes with high fidelity to approximate the target surfaces accurately. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive mesh-adjustment strategy for continuum topology optimization to achieve manufacturable structural layout

This paper presents an adaptive mesh adjustment algorithm for continuum topology optimization method to describe the structural boundary using nonuniform isoparametric element. A criterion on the basis of the node movement is proposed; herein, the densities and coordinates of the nodes are defined to instruct the deformation of finite elements in subsequent optimization iterations. With such a scheme, the topology optimization can start from a regular mesh discretization then gradually yields an optimal design with clear structural boundaries. The element in the transition along the boundary is refined; on the contrary, the pure solid or void element is coarsen. The contribution of this work is to improve the resolution of the structural boundaries and decrease the percentage of transitional regions with the invariant design variable. Several 2D and 3D numerical examples indicate the effectiveness of our proposed method. Seen from the examples, the structural boundary become smoother and the intermediate densities have been reduced up to 70%. In addition, a design process based on the presented method is proposed to make the optimum solutions be fabricated conveniently and accurately by linking it with the 3D design software, ie, SolidWorks, which is also demonstrated in the numerical examples.     

A scaled boundary finite element based explicit topology optimization approach for three-dimensional structures

This article proposes an efficient approach for solving three-dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)-based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.     

An imprint and merge algorithm incorporating geometric tolerances for conformal meshing of misaligned assemblies

A size tolerant algorithm for imprinting and merging adjacent part geometries is presented in this paper. This approach cleans up misaligned and dirty assembly geometry that is intended for conformal mesh generation. The algorithm first discretizes the boundary edges of adjacent faces into linear segments. The segments are then intersected and a partial intersection graph is calculated from the intersection results. The intersections are calculated by accounting for size tolerances to minimize the effect of misaligned parts on the meshing process. The partial intersection graph is next used to imprint the adjacent faces as appropriate. The imprints are generated using virtual geometry so that the tolerant topology created therein can be employed. Several examples are given to demonstrate the resulting improvement in the mesh quality of conformal meshes. The approach is shown to work robustly with misaligned and poorly defined parts. Copyright © 2004 John Wiley & Sons, Ltd.     

A consistent point‐searching algorithm for solution interpolation in unstructured meshes consisting of 4‐node bilinear quadrilateral elements

To translate and transfer solution data between two totally different meshes (i.e. mesh 1 and mesh 2), a consistent point‐searching algorithm for solution interpolation in unstructured meshes consisting of 4‐node bilinear quadrilateral elements is presented in this paper. The proposed algorithm has the following significant advantages: (1) The use of a point‐searching strategy allows a point in one mesh to be accurately related to an element (containing this point) in another mesh. Thus, to translate/transfer the solution of any particular point from mesh 2 to mesh 1, only one element in mesh 2 needs to be inversely mapped. This certainly minimizes the number of elements, to which the inverse mapping is applied. In this regard, the present algorithm is very effective and efficient. (2) Analytical solutions to the local co‐ordinates of any point in a four‐node quadrilateral element, which are derived in a rigorous mathematical manner in the context of this paper, make it possible to carry out an inverse mapping process very effectively and efficiently. (3) The use of consistent interpolation enables the interpolated solution to be compatible with an original solution and, therefore guarantees the interpolated solution of extremely high accuracy. After the mathematical formulations of the algorithm are presented, the algorithm is tested and validated through a challenging problem. The related results from the test problem have demonstrated the generality, accuracy, effectiveness, efficiency and robustness of the proposed consistent point‐searching algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

An effective 3D meshing approach for fractured rocks

An effective mesh generation algorithm is proposed to construct mesh representations for arbitrary fractures in 3D rock masses. With the development of advanced imaging techniques, fractures in a rock mass can be clearly captured by a high‐resolution 3D digital image but with a huge data set. To reduce the data size, corresponding mesh substitutes are required in both visualization and numerical analysis. Fractures in rocks are naturally complicated. They may meet at arbitrary angles at junctions, which could derive topological defects, geometric errors or local connectivity flaws on mesh models. A junction weight is proposed and applied to distinguish fracture junctions from surfaces by an adequate threshold. We take account of fracture junctions and generate an initial surface mesh by a simplified centroidal Voronoi diagram. To further repair the initial mesh, an innovative umbrella operation is designed and adopted to correct mesh topology structures and preserve junction geometry features. Constrained with the aforementioned surface mesh of fracture, a tetrahedral mesh is generated and substituted for the 3D image model to be involved in future numerical analysis. Finally, we take two fractured rock samples as application examples to demonstrate the usefulness and capability of the proposed meshing approach. Copyright © 2015 John Wiley & Sons, Ltd.     

大型飞机三维空间噪声的主动控制研究

三维空间的噪声主动控制技术是近年来噪声控制领域研究的一个热点问题,对于降低飞机或车辆里的噪声水平有很大意义。提出了一种对于噪声在三维空间里进行优化控制的方法,建立三维空间的几何控制模型,并且对几何模型进行区域划分,误差准则取所有区域的声压平方和,确定优化目标为误差准则取得最小值,计算出多点次级声源的强度。最后基于该方法设计的噪声主动控制系统进行数值仿真和实验验证,结果验证了该方法的正确性和有效性。     

An algorithm for stress and mixed control in Galerkin-based FFT homogenization

A new algorithm is proposed to impose a macroscopic stress or mixed stress/deformation gradient history in the context of nonlinear Galerkin-based fast Fourier transform homogenization. The method proposed is based on the definition of a modified projection operator in which the null frequencies enforce the type of control (stress or strain) for each component of either the macroscopic first Piola stress or the deformation gradient. The resulting problem is solved exactly as the original variational method, and it does not require additional iterations compared to the strain control version, neither in the linear iterative solver nor in the Newton scheme. The efficiency of the proposed method is demonstrated with a series of numerical examples, including a polycrystal and a particle-reinforced hyperelastic material.     

基于关节层次模型的3D手臂运动快速跟踪算法

针对人体3D运动跟踪过程中计算效率低下的问题,提出了基于粒子滤波和关节层次模型的3D手臂运动快速跟踪算法.该算法在经典的人体3D骨骼模型的基础上,提出了关节层次模型,利用关节层次模型将高维手臂运动状态参数空间分解为若干个低维参数子空间,最后,在关节层次模型的约束条件下,根据自顶向下的搜索策略,利用粒子滤波算法对手臂运动进行跟踪,从而减少了跟踪所需的粒子数目.实验表明,与标准的粒子滤波算法相比,该算法能够在保证跟踪精度的同时,提高计算效率.     

An indirect Boundary Element Method for solving low Reynolds number Navier–Stokes equations in a three-dimensional cavity

In the present work, we propose an indirect boundary-only integral equation approach for the numerical solution of the Navier–Stokes system of equations in a three-dimensional flow cavity. The formulation is based on an indirect integral representational formula for the permanent Stokes equations, and the use of a particular solution of a nonhomogeneous Stokes system of equations in order to obtain in an iterative way the corresponding complete solution of the problem. Previous boundary-only integral equation approaches to the present problem, using direct boundary elements formulations, result in a series of matrix multiplications that make these approaches computationally costly. Due to the use of an indirect formulation, the present approach is free from those matrix multiplications. © 1998 John Wiley & Sons, Ltd.