Finite element calculations

In order to describe inherited stress state of weldment the finite element modelling results are used. A series of finite element calculations were conducted to model step-by-step residual stresses as well as its redistribution due to heat treatment and operation [3]. The solutions for the reference weldment geometries are collected in the data base. If necessary (some variants of repair) the modelling is executed for this specific case. 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

Figure 21 shows the results from a finite-element calculation of the voltage from the cathode to various points in the anode for an anode design with an internal metal conductor as in Fig. 20. 

As mentioned in Section 21.4.1, the existence of intermediate phase with slightly stiffer modulus than that of mbber matrix was reported, which was determined by finite element calculation. The author reported that there are two phases around CB—one is almost comparable with bound mbber, 2 nm-thick glassy phase (GH-phase), another is 10 nm-thick uncross-Unked one (SH-phase). The intermediate regions in this smdy were usually observed around CB regions and the Young s modulus of this region was higher as well. Thus, there is a possibility that SH-phase was directly observed in real space for the first time. 

Figure 50. (a) Primary potential distribution surrounding a disk-shaped electrode. (Reprinted with permission from ref 47. Copyright 2004 John Wiley Sons.) (b) Finite-element calculation of the 2D primary potential profile near an electrode with regular periodic contact to the electrolyte

contour plot are lines of constant potential vertical lines follow the current path. 

Figure 54. Measured (a) and simulated (b) effect of electrode misalignment, (a) Total-cell and balf-cell impedances of a symmetric LSC/rare-earth-doped ceria/LSC cell with nominally identical porous LSC x= 0.4) electrodes, measured at 750 °C in air based on tbe cell geometry shown. (b) Finite-element calculation of tbe total-cell and half-cell impedances of a symmetric cell with identical R—C electrodes, assuming a misalignment of the two working electrodes (d) equal to the thickness of the electrolyte (L). ...

Fig. 9.12. Accuracy of the analytic expression. A comparison of Eq. (9.30) with the finite-element calculation by Carr (1988). (Reproduced from Chen, 1992, with permission.)...

Fig. 8.9. Stress fields at the end of a trench etched in a 15f

The highest grouping level in the hierarchy is called a domain. This term is commonly used in finite-element calculations to denote different regions of a problem where there may be different physical properties or governing equations. This is the sense in which we use the term. Because fundamentally different reaction chemistry may be occurring in two spatial regions, say in the gas and on a reactive surface, it is convenient to divide the calculation into domains. 

Fig. 5a-c Principal stress values for rectangular bars loaded along their length in tension by forces applied at the four corners. (Finite element calculations in plane strain), a Isotropic material aspect ratio 5 b Isotropic material aspect ratio 10 c Anisotropic material (E/G),n = 15.1, aspect ratio 10 (Note the figures are not drawn to scale). (From Ref. 16)... 

Fig. 6. Tensile strain along direction of loading at surface (top) and at centre (bottom) in a rectangular bar loaded along its length by forces at its four comers. (Finite element calculations in plane strain). Anisotropic material (E/G)1 2 = 15.1. Aspect ratio 40/3...

M.F. Horstemeyer et al Micromechanical finite element calculations of temperature and void configuration effects on void growth and coalescence. Int J. Plasticity 16, 979-1015 (2000)... 

Fig. 14. Hypothetical bi-grain with microelectrodes (a), and square electrodes on the sides (b). (d) Current density distribution in the grain boundary plane of the 24-grain model sample, and sketch of the grain boundaries for which the current density distribution is plotted (c). Parameters used in the finite element calculations dms = 1/5 Lg, wgb = 10 3 Lg, and p b = 105 phvSk for all grain boundaries. The spikes are due to numerical errors. Owing to computational reasons, the considered plane is not the grain boundary plane itself, but a plane in a distance of 0.052., to the grain boundaries.

Therefore let us instead consider the more practical case of the tertiary current distribution. Based on the dependency of the Wagner number on polarization slope, we would predict that a pipe cathodically protected to a current density near its mass transport limited cathodic current density would have a more uniform current distribution than a pipe operating under charge transfer control. Of course the cathodic current density cannot exceed the mass transport limited value at any location on the pipe, as said in Chapter 4. Consider a tube that is cathodically protected at its entrance with a zinc anode in neutral seawater (4). Since the oxygen reduction reaction is mass transport limited, the Wagner number is large for the cathodically protected pipe (Fig. 12a), and a relatively uniform current distribution is predicted. However, if the solution conductivity is lowered, the current distribution will become less uniform. Finite element calculations and experimental confirmations (Fig. 12b) confirm the qualitative results of the Wagner number (4). 

No attempt has been made to discuss, in a comprehensive manner, models which are based on finite element calculations or other numerical analyses. Only some results of Schmauder and McMeeking10 for transverse creep of power-law materials were discussed. The main reason that such analyses were, in general, omitted, is that they tend to be in the literature for a small number of specific problems and little has been done to provide comprehensive results for the range of parameters which would be technologically interesting, i.e., volume fractions of reinforcements from zero to 60%, reinforcement aspect ratios from 1 to 106, etc. Attention in this chapter was restricted to cases where comprehensive results could be stated. In almost all cases, this means that only approximate models were available for use. 

Errors are a result of the elastic and plastic nature of the ceramic particles themselves. Several other authors [81,82] have calculated the pressure distribution in cylindrical dies and other forms using finite element numerical methods. Bortzmeyer [81] has incorporated cohesion, elastic, and plastic deformation of the particles into finite element calculations for more complicated geometries, as shown in... 

The effect of different types of interlayer on thermoelastic residual stresses can be analyzed from finite-element calculations for a two-dimensional geometry, assuming perfect adherence and without taking into account any reactivity between the components. 

The first part of the paper is devoted to the design of the notched specimen and the analysis of the resulting stress intensity factor fi om a finite element calculation. The second part deals with the practical preparation of the samples. In a third part, the evolution of the toughness and the analysis of the crack tip fields with the loading rate are presented for PMMA and PC. The influence of the notch radius is also considered. 

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the G1 un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. 

Fig.2. Comparison of the SIF of a single notch under pure bending (Ki ) and that from finite element calculation of the unotched G1 configuration (Ki). The different x values represent increasing off centre, normalised by the inner span.

It was found that formula (6) leads to better results even for very short cracks. In this comparative theoretical analysis, the deflection d in formulas (5)-(7) is taken from finite-element calculations. 

In the present paper the Boundary Finite Element Method is presented as a boundary discretization method for the numerical investigation of interfacial stress concentrations in composite laminates. In contrast to the classical boundary element method, the element formulation is finite element based, which avoids the necessity of a fundamental solution. Comparative results from finite element calculations show good agreement both for the laminate free-edge effect and for the example of the stress concentrations near cracks in composite laminates. 

Fig. 3.8. Two different eigenfunctions for the quantum corral as obtained by using a two-dimensional finite element calculation (courtesy of Harley Johnson).

Numerical Calculation of Two-Dimensional Equilibrium Shapes. To go beyond the relative simplicity of the set of shapes considered in the analysis presented above it is necessary to resort to numerical procedures. As a preliminary to the numerical results that will be considered below and which are required when facing the full complexity of both arbitrary shape variations and full elastic anisotropy, we note a series of finite-element calculations that have been done (Jog et al. 2000) for a wider class of geometries than those considered by Johnson and Cahn. The analysis presented above was predicated on the ability to extract analytic descriptions of both the interfacial and elastic energies for a restricted class of geometries. More general geometries resist analytic description, and thus the elastic part of the problem (at the very least) must be solved by recourse to numerical methods. 

As a complementary approach a mean-field method is used in combination with the finite element method to investigate the FGM. To compare the predictions with the periodic unit cell simulations, the FGM part is divided into nine sublayers. Each of them consists of two bi-quadratic 8-node plane elements over the thickness, and only one element is used in the horizontal direction. The parts of pure alumina and pure nickel are modeled by three and 12 elements, respectively. In each sublayer the volume fractions of the phases are constant. In addition, the center sublayer can be split into a metal-matrix and a ceramic-matrix half sublayer. The properties of the particular material on the meso-structural level within the finite element calculation are described via a con-... 

The periodic unit cell results are directly comparable to the IMT predictions, because both approaches represent the same matrix/inclusion type microstructure. However, such comparisons have to be done carefully since some assumptions regarding the finite element calculations are not equivalent for the extended unit cell approaches and the present mean-field method. The plane stress analysis of the unit cell models does not take into account the constraints in the out-of-plane direction. In contrast, within the present IMT formulation the inclusions are enclosed three-dimensionally by the matrix material. In contrast to the plane stress unit cell models, the constraint in the out-of plane direction is accounted for. Accordingly, these predictions are denoted as full internal constraint. To overcome this internal constraint in order to simulate the plane stress model assump-... 

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