Fig. 7. Pressure vs distance plot for the continuous pressure formulation of  and for the discontinuous pressure formulation presented here. The continuous and discontinuous pressure solutions coincide within the plotted line width.
Fig. 8. (A) Permeability of the  test model with one-element thick barriers (shown in red). The zones in orange have an isotropic dimensionless perme- ability of 100; the one-element thick barriers have an isotropic dimensionless permeability of 10 − 4 . The regions in blue have an anisotropic dimensionless permeability of 900 and 100 oriented as shown. (B) Snapshot of the injected wetting phase saturation obtained by  using a conventional CVFE method in which CVs span several FEs. The non-wetting phase is injected through the lower left corner and ﬂuid is extracted from the right top corner. Note how the non-wetting phase is transported through the low-permeable barriers; this is a numerical artifact introduced by the control volumes that span the boundary between the high and low permeability domains. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)
Staggered meshes, in which different variables are evaluated in different points of the computational mesh (some at the cell centers, others at the cell faces), were used by Yoo and Na (1990), as well as in subsequent works (eg. Gervang and Larsen, 1991; Sasmal, 1995; Xue et al, 1995, 1998 a,b; Mompeam and Deville, 1997; Bevis et al, 1992). Staggered meshes provide an easy way to couple velocities, pressure and stresses, but calculations involving complex geometries become rather difficult and in some cases do not allow for the determination of the shear stress at singular points, such as re-entrant corners. Alternatively, the use of non-orthogonal, or even non-structured meshes, are to be preferred in such cases. Non-orthogonal meshes have been used in FVM for Newtonian fluids since the mid- nineteen eighties, but its application to finitevolume viscoelastic methods happened only in 1995. Initially, the adaptation to computational rheology of some of the techniques previously developed for Newtonian fluids to has been slow, namely on issues like pressure-velocity coupling for collocated meshes (in which all variables are evaluated at the cell centres), time marching algorithms or the use of non-orthogonal meshes. Lately, progress has been quicker on issues of stability for convection dominated flows, as in viscoelastic flows at high Weissenberg numbers and in high speed flows of inviscid Newtonian fluids involving shock waves (Morton and Paisley, 1989; Mackenzie et al, 1993). Regarding other mesh arrangements, Huang et al (1996) used non-structured methods in a mixed finiteelement/finitevolume formulation by extending the controlvolumefiniteelementmethod (CVFEM) of Baliga and Patankar (1983) for the prediction of the journal bearing flow of Phan-Thien and Tanner (PTT) fluids. Nevertheless, the formulation lacked the generality of modern methods in Newtonian fluid calculations on collocated grids (Ferziger and Perić, 2002) and was problematic to extend to higher-order shape functions (usually the convective terms are discretized with some form of upwind). Later, Oliveira et al (1998) developed a general method for solving the full momentum and constitutive equations on collocated non-orthogonal meshes, enabling calculations of complex three-dimensional flows. Their scheme for coupling velocity, pressure and stresses was later improved by Oliveira and Pinho (1999a) and Matos et al (2009). This issue was also addressed in a parallel effort by Missirlis et al (1998), but only for staggered, orthogonal meshes.
Incorporating nonlocal theories into numerical methods for predicting the behaviour of loaded heterogeneous materials is not as straightforward as with classical elasticity theory. Gradient theories ideally require inter element continuity of higher order displacement derivatives and this is not generally exhibited by conventional finiteelement (FE) methods. The development of suitable elements offering this feature is actively being pursued [3, 4, 5]. Micropolar elasticity theory has been incorporated into FE methods by a number of researchers [6, 7, 8, 9, 10]. However, the numerical results reported indicate that there are some issues that arise. Firstly, the accuracy of the numerical solutions can vary as the values of the additional micropolar elasticity parameters are changed even though the mesh remains unaltered [7, 8, 10]. Secondly, some elements appear to only satisfy appropriate patch tests approximately rather than exactly as might be expected . Finally, reported results invariably consider two dimensional cases only implying that full three dimensional analyses are an as yet unrealized challenge for FE methods.
In this section we introduce a variation of the ﬁnite elementmethod (FEM). The diﬀerence from this method is the introduction of a dual mesh, and that we approximate it with the ﬂux over the so called control volumes generated around every node instead of using the original triangulation to calculate the solution. For more about the FEM see [3, 7]. In the following chapter we will study the essence of the controlvolume ﬁnite elementmethod. Basic concepts of the method will be explained. First important deﬁnitions and syntax such as mesh, basis, shape functions, region of support, controlvolume will be explained before we explain the method through a model problem. In later chapters, important aspects of implementation will be discussed. A fundamental part of the CVFEM is the discretization and how this is applied. The ﬁrst part of this chapter will be dedicated to how we prepare the continuous problem to suit the discrete version of the method. Much of the theory about CVFEM shown in this chapter are inspired from [21, 9, 7, 13].
of which can be seen in table 3. The internal vertex nodal coordinates and constitutive properties can be found in figure 7. The plane strain formulation was used. The first patch is for a uniform direct stress with symmetric shear. In the second test the direct stress remains uniform whereas the shear stress is now asymmetric and a body couple is applied. The final test has constant direct stresses and body forces, linearly varying body couples and linearly varying asymmetric shear. The controlvolumemethod CV-MPLST detailed here passes the first two tests, table 4, while results for the final test are shown in table 5 where a comparison is made with the earlier constant strain controlvolume, CV-MPCST, which has been shown to out perform the equivalent, constant and linear strain, finiteelement formulations . As can be seen, the CV-MPLST does not appear to reproduce the analytical solution exactly, unlike CV-MPCST formulation, but the differences are so small they are in all likelyhood attributed to rounding error.
The finiteelementmethod (FEM) is a numerical method of analysis for stresses and deformations in structures of any given geometry. The structure is discretized into the so called ‘finite elements’ connected through nodes. The type, arrangement and total number of elements affect the accuracy of the results. The FEM has become one of the most successful engineering computational methods and most useful analysis tool since the 1960s. 4,5 It is showing overwhelming capability and versatility in its application in dentistry. 6-23 This paper reviews past and current practices in the finiteelement analysis of dental implants. The achievements and limitations of the existing analysis are discussed and the gap in research is identified. Future research directions are also recommended with particular emphasis on the stress evaluation and design optimisation associated with the implants.
6.2 Closing Remarks
The virtual elementmethod has opened a whole new world of interesting prob- lems within numerical mathematics, both theoretical, and technical. One of the most interesting features of VEM is the stability term, or rather the freedom in choosing it. We have seen that it is possible to obtain both finiteelement and finite difference discretizations from suitable choices of this term, and it is rea- sonable to assume that it is possible to obtain several other methods from VEM as well. Such equivalences are of particular interest, since we then know that the properties of these methods also applies to VEM. Moreover, the equivalence might provide useful information about the other method based on the existing theory for VEM, and give a natural way to extend the method to more general cell geometries.
This paper described two possible strategies of solution of an classical aeroelastic problem. The first on based on the classical approach of linearized aerodynamics was susccessfully applied for solution of two aeroelastic prob- lems from references. Its results were also compared to the other method described in this paper, which is the fi- nite element approximations of the FSI problem. Both ap- proaches were found to be very similar in terms of the pre- dicted frequency and damping for sub-critical velocities. However for far field velocities close to the critical one it was found that the latter approach based on the finite ele- ment method can be influenced by the nonlinearities pre- sented in the system as large displacements, high initial displacement etc.
conditions, hence to the establishment of the FE formulation of the problem. The plate elements developed are the two-dimensional triangular element. To meet the convergence criteria, the quadratic interpolation function is adopted and the six nodes triangular element is developed. The deflection w takes the form of
3.8. An optimal control problem with Neumann boundary condition. In this section solve finiteelement equations as one equation, this can be done by adding all finiteelement equations then it is an equation which contain more than one variables. In beginning solve individually. Then solve two stationary point equations as one equation. Finally solve all three finiteelement equations as one equation. To solve finiteelement equations as one equation choose the objective function subjected to elliptic PDE’s constraints with Neumann boundary conditions defined as:
Abstract: Analysis of perforated plate by finiteelementmethod is the subject of present paper. Hinton's eight noded isoparametric plate element based on thick plate theory, is utilised for the analysis. The plate theory, in terms of transverse deflection and rotations of the normal about X and Y axes, involves only first derivative in energy expression and hence only C o continuity is needed in shape functions. The bending, twisting and shear energies are
In the mesh partition step, the METIS applied for nodes partition is a set of serial pro- grams for partitioning graphs and partitioning finiteelement meshes. CSMP firstly use meshes partitioning program in METIS, to obtain both mesh elements partition and nodes partition. Secondly, CSMP builds sub finiteelement meshes for each nodes partition, the criterion for such element is that if all nodes of an element belongs to a partition, then the nodes will be assigned to that partition. Thirdly, CSMP complete each sub mesh with its halo elements, the criterion for such halo element is that if and only if part of the nodes of an element belongs to a partition, then it will become a halo element to that partition, and therefore a halo element thus will be in at least two sub meshes. In this halo elements step, halo nodes will also be set for communication, halo nodes are the nodes in the halo elements. Finally, CSMP creates finite volumes for each node in all of the partitioned meshes. Figure 1.4 could help explain these operations. Currently, the problem with the mesh partition is that METIS can not work with hybrid element mesh partition, which prevent CSMP from obtaining more benefit of combining hybrid element mesh and parallelization. This issue will be detailed in Section 1.3. SAMGp used in the parallel matrix solving step is a commercial library for solving domain partition based matrix, and its corresponding serial version SAMG  is also used in the serial CSMP matrix solving. The commercial SAMG uses a complicated licensing system named FLEXnet to manage its usage. In this project, correctly setting up and running SAMG on the target machine is a crucial step for running CSMP and any further development.
Figure 2.1 (c) Change in volume with no change in shape 6 Figure 3.1 FiniteElement Discretisations of a machine component 21 Figure 3.2 Finite Difference Discretisations of a machine component 22 Figure 3.3 Boundary Element Discretisations of a machine component 23 Figure 3.4 The illustration 2-noded bar element 23 Figure 3.5 The iilustration for torsion problem 26 Figure 3.6 The illustration for torsion problem for element 1 27
Another approach is to numerically solve bulk equations in one space dimension higher. This may be a natural approach when the surface is computed implicitly using phase ﬁeld or level set methods [4, 32, 29]. The idea is then to exploit the im- plicit formulation and use a bulk triangulation rather than a surface triangulation. One idea is to solve the surface partial diﬀerential equations on all level sets of a prescribed function. This is inherently an Eulerian method and yields degenerate equations; see [2, 19, 18, 11] for stationary surfaces. For surface elliptic equations,  gave a discretisation error analysis for a narrow band level set method using the unﬁtted ﬁnite elementmethod. Another method using a bulk unﬁtted mesh and ﬁnite element space independent of the surface which is given by a level set function has been proposed in [27, 28]. Eulerian approaches to transport and diﬀusion on evolving surfaces were given in [1, 33] where level set approximations to surface quantities were required. On the other hand, an elegant formulation avoiding the need to do this was provided in  using an implicit surface version of the Leibniz formula.
Abstract During the past two decades, a large variety of mesh-free methods have been introduced as superior alternatives to the traditional FEM. However, the acceptance in professional practices seems to be slow due to their implementation complexities. Recently, the authors proposed a very convenient implementation of Element-free Galerkin Method (EFGM) using the node-based Kriging interpolation (KI). Two key properties of KI are Kronecker delta and consistency properties. Due to the former, KI passes through all the nodes thus requiring no special treatment for boundary conditions. The consequence of the latter ensures reproduction of a linear interpolation if the basis function includes the constants and linear terms. In this study, layers of finite elements around any node are adopted as its domain of influence. This method is referred to as Kriging-based FEM (K-FEM), which can be viewed as a generalized form of FEM. Precisely, if we limit the nodal domain of influence to only one finiteelement layer around the node, K-FEM specializes to the traditional FEM.
from 200 kHz to 1 MHz and the laser beam temperature from 100 ◦ C to 500 ◦ C. The diﬀusion of heat from laser beam and the eddy current density distribution around the crack are shown in Fig. 4. The temperature ﬂuctuation causes a local change in the electrical conductivity of the specimen and the current density of the specimen. The lines indicate the contour of induced current density on the coil and the specimen. Figs. 5 and 6 show the signal of coil impedances with EC method and PI method, respectively. The center point of the rectangular notch is 0 mm in x-axis, as shown in Figs. 5 and 6. Fig. 5 is the EC image signals of a 0.5-mm rectangular notch at 600-kHz EC frequency, without laser beam. Fig. 6 is the PI image signals of a 0.5-
The challenges for convergence and stability analysis of discretization methods for advection-diﬀusion equations on moving surfaces arise because of the moving ge- ometry. Existing results are for direct approximations based on interpolations of the evolving surface. An optimal order error analysis for the semidiscrete ESFEM based on piecewise linear ﬁnite element approximations has been given in [4, 6]. High or- der Runge–Kutta schemes for this semidiscretization have been derived and analyzed in . In particular, stability and optimal order error bounds for the ordinary dif- ferential equation systems arising from ESFEM approximation of advection-diﬀusion equations on moving surfaces are obtained. In  Lenz, Nemadjieu, and Rumpf proved L 2 and H 1 error bounds for their proposed fully discrete time implicit ﬁnite volume scheme. The purpose of this paper is to extend the results of  to the case when the time discretization is based on a backward Euler scheme.
C continuity, but the former with a general polynomial degree (Lagrange polynomials) and the latter with well-known linear and quadratic functions. Fig. 4b shows that increasing the polynomial order inside each element the error increases. In fact the SFEM error with N 7 is higher than the FEM one with N 2 . Nevertheless, the maximum error is always below the machine working precision (black dashed line). A comparison in terms of dynamic discrete spectra in shown in Fig. 5a. It can be noticed that SFEM offers the best accuracy in terms of percentage of accurate modes and the maximum errors are always more limited than all the other techniques. Fig. 5b shows several convergence rates with respect to the first natural frequency. Obviously, increasing the polynomial degree the convergence rate passes from 1/2 to 1/10. Linear FEM ( N 2 ) has the same trend (1/2) of the ones obtained by SFEM and WFEM with N 3 and 1