Numerical and Experimental Models of the Mandible

This study aimed to validate a numerical model of an intact mandible for further development of a new TMJ implant. Numerical and experimental models of the biomechanics of the mandible were elaborated to characterize the human temporomandibular joint and to approach the development of a condyle implant. The model of the mandible was obtained through the use of a polymeric replica of a human cadaveric mandible and through 3D geometry acquisition. The three-dimensional finite element model was generated as a tetrahedral finite element mesh. The level of mesh refinement was established via a convergence test and a model with more than 50,000 degrees of freedom was required to obtain analysis accuracy. The functional loading cases included muscle loading in four different load boundary conditions. The same boundary conditions were applied to the experimental model. The strains were measured with an experimental procedure using electric resistance strain gauges applied on the external surface of the mandible. The mechanical response is shown and discussed in terms of strains, principal numerical and measured strains. This study proved that FE models of the mandible can reproduce experimental strains within an overall agreement of 10%. The FE models correctly reproduced bone strains under different load configurations and therefore can be used for the design of a novel TMJ implant considering other load configurations and bone mechanical properties.     

Error sensitivity to refinement: a criterion for optimal grid adaptation

Most indicators used for automatic grid refinement are suboptimal, in the sense that they do not really minimize the global solution error. This paper concerns with a new indicator, related to the sensitivity map of global stability problems, suitable for an optimal grid refinement that minimizes the global solution error. The new criterion is derived from the properties of the adjoint operator and provides a map of the sensitivity of the global error (or its estimate) to a local mesh refinement. Examples are presented for both a scalar partial differential equation and for the system of Navier–Stokes equations. In the last case, we also present a grid-adaptation algorithm based on the new estimator and on the \(FreeFem++\) software that improves the accuracy of the solution of almost two order of magnitude by redistributing the nodes of the initial computational mesh.     

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

In this paper, we present an approach of dynamic mesh adaptation for simulating complex 3‐dimensional incompressible moving‐boundary flows by immersed boundary methods. Tetrahedral meshes are adapted by a hierarchical refining/coarsening algorithm. Regular refinement is accomplished by dividing 1 tetrahedron into 8 subcells, and irregular refinement is only for eliminating the hanging points. Merging the 8 subcells obtained by regular refinement, the mesh is coarsened. With hierarchical refining/coarsening, mesh adaptivity can be achieved by adjusting the mesh only 1 time for each adaptation period. The level difference between 2 neighboring cells never exceeds 1, and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases. A predictor‐corrector scheme is introduced to eliminate the phase lag between adapted mesh and unsteady solution. The error caused by each solution transferring from the old mesh to the new adapted one is small because most of the nodes on the 2 meshes are coincident. An immersed boundary method named local domain‐free discretization is employed to solve the flow equations. Several numerical experiments have been conducted for 3‐dimensional incompressible moving‐boundary flows. By using the present approach, the number of mesh nodes is reduced greatly while the accuracy of solution can be preserved.     

A high‐order element based adaptive mesh refinement strategy for three‐dimensional unstructured grid

Adaptive mesh refinement (AMR) shows attractive properties in automatically refining the flow region of interest, and with AMR, better prediction can be obtained with much less labor work and cost compared to manually remeshing or the global mesh refinement. Cartesian AMR is well established; however, AMR on hybrid unstructured mesh, which is heavily used in the high‐Reynolds number flow simulation, is less matured and existing methods may result in degraded mesh quality, which mostly happens in the boundary layer or near the sharp geometric features. User intervention or additional constraints, such as freezing all boundary layer elements or refining the whole boundary layer, are required to assist the refinement process. In this work, a novel AMR strategy is developed to handle existing difficulties. In the new method, high‐order unstructured elements are first generated based on the baseline mesh; then the refinement is conducted in the parametric space; at last, the mesh suitable for the solver is output. Generating refined elements in the parametric space with high‐order elements is the key of this method and this helps to guarantee both the accuracy and robustness. With the current method, 3‐dimensional hybrid unstructured mesh of huge size and complex geometry can be automatically refined, without user intervention nor additional constraints. With test cases including the 2‐dimensional airfoil and 3‐dimensional full aircraft, the current AMR method proves to be accurate, simple, and robust.     

A new analytical model for the elastic–plastic behaviour of spot welded joints subjected to orthogonal load

An analytical procedure for the evaluation of the elastic–plastic stiffness behaviour of spot welded joints is presented. The procedure is based on a new model of spot weld region: a circular plate having variable thickness with a central rigid nugget, which is resolved using an original analytical method.The closed-form solution allows to describe the displacement of a rigid nugget when an axial orthogonal load is applied on the plate while plasticity and large deflections are present. The goal is to reach a reliable spot weld region model which can be used as the basis to develop a spot weld element in FE analysis even when plasticity and large deflections are in effect.The procedure is as completely original as no other can be found in the technical literature, and it has been applied to some examples of plates usually employed for spot weld analysis. The analytical results obtained by using the new general relations precisely match those obtained modelling spot weld area by FEA.     

NESTED GRID METHODS FOR AN OCEAN MODEL: A COMPARATIVE STUDY

In this paper a comparison is carried out between three correction methods for multigrid local mesh refinement in oceanic applications: FIC, LDC and the direct method (DM) proposed by Spall and Holland. This study is based on a nested primitive equation model developed by Laugier on the basis of the code OPA (LODYC). The external barotropic problem is solved using any of the three local grid correction algorithms yielding an interactive nested grid model. The non-linear elliptic equation for the barotropic streamfunction tendency is solved on two nested grids, called the global and the zoom grid, that interact between themselves. The zoom grid is entirely embedded within the global domain with a horizontal grid step ratio of 3:1. The computation on the global grid supplies the boundary conditions for the zoom grid region and the fine grid fields are used to correct the global coarse solution. The three local correction methods are tested on two problems relevant to oceanic circulation phenomena proposed by Spall and Holland: a barotropic modon and an anticyclonic vortex. The results show that the nesting technique is a very efficient way to solve these problems in terms of a gain in precision compared with the required CPU time. The two-domain model with local mesh refinement allows one both to manage effectively the open boundary conditions for the local grid and to correct the global solution thanks to the zoom solution. In the case of the modon propagation the three local correction methods provide approximately the same results. For the baroclinic vortex it appears that the two iterative methods are more efficient than the direct one.     

On the stability of integral equations of the first kind with logarithmic kernels

This paper discusses the stability of the Galerkin method for a class of boundary integral equations of the first kind. These integral equations arise in acoustics, elasticity, and hydrodynamics, and the kernels of the principal parts of the corresponding integral operators all have logarithmic singularities. It is shown that an optimal choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence of the approximate solutions. The results here are consistent with those obtained by the Tikhonov regularization procedure.     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

薄板振动特征值问题的一种改进积分方程解法

本文根据一种改进的边界元/有限元混合法求解薄板振动固有频率问题,既避开了标准的边界元法所导致的求解非代数特征值方程的困难,亦能够基本上消除通常的边界元/有限元混合法结果精度受区域内部单元划分影响较大的弊端。文中讨论了迭代算法的收敛问题,并用于薄板固有频率分析。数值结果表明,即便是在域内单元很粗疏划分的情况下,本文的方法仍能给出相当满意的结果。     

Sequential feature‐based mesh movement and adjoint error‐based mesh refinement

Nowadays, aerodynamic computational modeling is carried out on a daily basis in an industrial setting. This is done with the aim of predicting the performance and flow characteristics of new components. However, limited resources in terms of time and hardware force the engineer to employ relatively coarse computational grids, thus achieving results with variable degree of inaccuracy. In this article, a novel combination of feature and adjoint‐based mesh adaptation methods is investigated and applied to typical three‐dimensional turbomachinery cases, such as compressor and fan blades. The proposed process starts by employing feature‐based mesh movement to improve the global flow solution and then adjoint refinement to tune the mesh for each quantity of interest. Comparison of this process with one utilizing only the adjoint refinement procedure shows significant benefits in terms of accuracy of the performance quantity.     

基于非局部塑性模型的应变局部化理论分析及数值模拟

通过求解一个第二类Fredholm方程,得到了基于非局部塑性软化模型的应变局部化问题理论解,结果表明,只有在当采用过非局部修正形式的非局部塑性软化模型才能得到应变局部化解,且得到的塑性应变分布和荷载响应依赖于所引入的特征长度及过非局部权参数。通过一维应变局部化有限元数值解,验证了非局部理论的引入能克服计算结果的网格敏感...     

Direct Extraction of Cohesive Fracture Properties from Digital Image Correlation: A Hybrid Inverse Technique

The accuracy of an adopted cohesive zone model (CZM) can affect the simulated fracture response significantly. The CZM has been usually obtained using global experimental response, e.g., load versus either crack opening displacement or load-line displacement. Apparently, deduction of a local material property from a global response does not provide full confidence of the adopted model. The difficulties are: (1) fundamentally, stress cannot be measured directly and the cohesive stress distribution is non-uniform; (2) accurate measurement of the full crack profile (crack opening displacement at every point) is experimentally difficult to obtain. An attractive feature of digital image correlation (DIC) is that it allows relatively accurate measurement of the whole displacement field on a flat surface. It has been utilized to measure the mode I traction-separation relation. A hybrid inverse method based on combined use of DIC and finite element method is used in this study to compute the cohesive properties of a ductile adhesive, Devcon Plastic Welder II, and a quasi-brittle plastic, G-10/FR4 Garolite. Fracture tests were conducted on single edge-notched beam specimens (SENB) under four-point bending. A full-field DIC algorithm was employed to compute the smooth and continuous displacement field, which is then used as input to a finite element model for inverse analysis through an optimization procedure. The unknown CZM is constructed using a flexible B-spline without any “a priori” assumption on the shape. The inversely computed CZMs for both materials yield consistent results. Finally, the computed CZMs are verified through fracture simulation, which shows good experimental agreement.     

AN ALGEBRAIC MULTIGRID SOLVER FOR NAVIER–STOKES PROBLEMS IN DISCRETE SECOND–ORDER APPROXIMATION

An algebraic multigrid scheme is presented for solving the discrete Navier–Stokes equations to second-order accuracy using the defect correction method. Solutions for the driven cavity and asymmetric, sudden expansion test problems have been obtained for both structured and unstructured meshes, the resolution and resolution grading being controlled by global and local mesh refinements. The solver is efficient and robust to the extent that, for problems attempted so far, no underrelaxation of variables has been required to ensure convergence. Provided that the computational mesh can resolve the flow field, convergence characteristics are almost mesh-independent. Rates of convergence actually improve with refinement, asymptotically approaching mesh-independent values. For extremely coarse meshes, where dispersive truncation errors would be expected to prevent convergence (or even induce divergence), solutions can still be obtained by using explicit underrelaxation in the iterative cycle.     

几何精确NURBS有限元中边界条件施加方式对精度影响的三维计算分析

非均匀有理B样条(NURBS)有限元法把计算机辅助几何设计(CAGD)中的NURBS几何构形方法与有限元方法有机结合起来,有效消除了有限元离散模型的几何误差,提高了计算精度。但是由于NURBS基函数不是插值函数,直接在控制节点上施加位移边界条件会引起较大误差。本文详细讨论了NURBS基函数的插值特性,在NURBS有限元分析中采用罚函数法施加位移边界条件,提高了收敛率和计算精度。结合典型三维弹性力学问题,对两种施加位移边界条件的方法进行了对比和分析。计算结果表明,直接施加位移边界条件会导致收敛率和精度的明显降低,而基于罚函数法的NURBS有限元分析则能达到最优收敛率,并具有更高的精度。     

Numerical simulations of the steady Navier–Stokes equations using adaptive meshing schemes

In this paper, we consider an adaptive meshing scheme for solution of the steady incompressible Navier–Stokes equations by finite element discretization. The mesh refinement and optimization are performed based on an algorithm that combines the so‐called conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) and residual‐type local a posteriori error estimators. Numerical experiments in the two‐dimensional space for various examples are presented with quadratic finite elements used for the velocity field and linear finite elements for the pressure. The results show that our meshing scheme can equally distribute the errors over all elements in some optimal way and keep the triangles very well shaped as well at all levels of refinement. In addition, the convergence rates achieved are close to the best obtainable. Extension of this approach to three‐dimensional cases is also discussed and the main challenge is the efficient implementation of three‐dimensional CfCVDT generation that is still under development. Copyright © 2007 John Wiley & Sons, Ltd.     

Adjoint‐based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods

We present a robust and efficient target‐based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection–diffusion problems, including the compressible Euler and Navier–Stokes equations. The hybridization of finite element discretizations has the main advantage that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the computational mesh. Consequently, solving for these DOFs involves the solution of a potentially much smaller system. This not only reduces storage requirements but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint‐based mesh adaptation over uniform and residual‐based mesh refinement and secondly, to investigate the efficiency of the global error estimate. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical solution of steady free‐surface flows by the adjoint optimal shape design method

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free‐boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. In this work we investigate the suitability of the adjoint shape optimization method for solving steady free‐surface flows. The asymptotic convergence behaviour of the method is determined for free‐surface flows in 2D and 3D. It is shown that the convergence behaviour depends sensitively on the occurrence of critical modes. The convergence behaviour is moreover shown to be mesh‐width independent, provided that proper preconditioning is applied. Numerical results are presented for 2D flow over an obstacle in a channel. The observed convergence behaviour is indeed mesh‐width independent and conform the derived asymptotic estimates. Copyright © 2003 John Wiley & Sons, Ltd.     

一种新的求解对流占优问题的自适应网格细化方法

在均匀网格上求解对流占优问题时,往往会产生数值震荡现象,因此需要局部加密网格来提高解的精度。针对对流占优问题,设计了一种新的自适应网格细化算法。该方法采用流线迎风SUPG(Petrov-Galerkin)格式求解对流占优问题,定义了网格尺寸并通过后验误差估计子修正来指导自适应网格细化,以泡泡型局部网格生成算法BLMG为网格生成器,通过模拟泡泡在区域中的运动得到了高质量的点集。与其他自适应网格细化方法相比,该方法可在同一框架内实现网格的细化和粗化,同时在所有细化层得到了高质量的网格。数值算例结果表明,该方法在求解对流占优问题时具有更高的数值精度和更好的收敛性。     

Mesh-Based Digital Image Correlation Method Using Non-Uniform Elements for Measuring Displacement Fields with High Gradient

A new mesh-based digital image correlation (mesh-DIC) scheme using non-uniform 8-node quadrilateral (Q8) elements is proposed to process displacement fields with regions of both high and low gradient. The effect of element size on Q8-DIC is extensively analyzed, and the results show that when the shape function of an element matches or over-matches the gradient of the displacement field, a larger element size can achieve better results. Conversely, when the shape function under-matches the gradient of the displacement field, a suitable element size should be chosen. Furthermore, a practical and effective method based on the residual error of the DIC calculation is proposed to automatically refine mesh with non-uniform elements. The effectiveness of this method is verified using simulated speckle images that contain one- and two-dimensional non-uniform displacements. The proposed method is also used to improve the accuracy of two experiments that measure displacement fields with high gradient.