Finite-element discretization

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

Weighted residual statements in the context of finite element discretizations... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

After application of the 6 time-stepping method (see Chapter 2, Section 2.5) and following the procedure outlined in Chapter 2, Section 2.4, a functional representing the sum of the squares of the approximation error generated by the finite element discretization of Equation (4.118) is formulated as... 

Figure 5.11 Finite element discretization of the symmetric domain in example 5.3.1...

The Galerkln finite element discretization leads to a set of nonlinear algebraic equations of the general form ... 

Figure 9.4 Schematic of a ID domain finite element discretization with 4 linear elements and 5 nodal points.

Figure 9.6 Simplified model of the Couette device of Fig. 9.5 with its finite element discretization.

The heat equation (Eq. 29) is coupled to the equations governing the mechanical response through the temperature dependence of the bulk viscoplastic strain rate (Eq. 3), the craze thickening rate (Eq. 22), and the thermal expansion in Eq. 1. The system of differential equations resulting from the finite element discretization of the energy balance in [9] is modified [57] to... 

Investigation 10 was a study of fixed-bed reactor models and their application to the data of Hettinger et al. (1955) on catalytic reforming of C7 hydrocarbons. The heuristic posterior density function p 6 Y) proposed by Stewart (1987) was used to estimate the rate and equilibrium parameters of various reaction schemes, two of which were reported in the article. The data were analyzed with and without models for the intraparticle and boundary-layer transport. The detailed transport model led to a two-dimensional differential-algebraic equation system, which was solved via finite-element discretization in the reactor radial coordinate and... 

Fig. 1 Considered part of the continuum with boundary finite element discretization...

On the other hand the relationship between the loads R and the displacements u at the discrete degrees of freedom at the nodal locations of the finite element discretization can be given by means of an elastic stiffness matrix K°°, which is independent from the considered cross-section ... 

In the finite-element discretization, the sub.script i denotes the ith finite element, and the subscript j (or k) denotes the yth (or /cth) collocation point in any finite element. There are a total of N finite elements and K collocation points (i = 1, 7= 1, A. The state variable X is approximated over each finite element by Lagrange interpolation basis functions (L/.(a)) as ... 

Fig. 16. Finite-element discretization for reaction-separation targeting model.

Carlone et al. (2006) examined using finite-difference and finite-element discretization methods to model the process with respect to cure and thermal properties, and found very good agreement between these models and experimental data Irom the litemture as found in Figure 6.8. 

If initial molecule concentrations are specified together with the concentrations at the compartment boundaries, RDEs can be solved numerically using finite-difference or finite-element discretizations of the compartment volume. The solution consists of molecular concentrations across the discretized compartment volume in a series of time points. RDEs and variants thereof have been used in a range of applications, including studies of pattern formation [Meinhardt and Gierer 2000 Myasnikova et al. 2001] and calcium transport [Smith, Pearson, and Keizer 2002]. 

Keywords Ionic polymer gels Modelling Numerical simulation Chemical stimulation Electrical stimulation Multi-field formulation Finite elements Discrete elements... 

Abstract A newly developed numerical simulator of two-phase flow using three-dimensional finite element method is presented in this paper. It is described that the fundamental simultaneous equations, the deduction to implicit pressure explicit saturation formulation and their finite element discretization method. Furthermore, its practical application to the numerical simulation project of predicting Horonobe natural gas product is also introduced. 

In this work, the commercial software package Comsol Multiphysics 3.2 (Corn-sol AB, Stockholm, Sweden) was used to solve the system of nonlinear partial differential equations. Apart from predefined built-in applications, this tool enables the user to define and solve PDE systems by finite-element discretization. Taking advantage of the axial symmetry of the problem, the model domain can be simplified to two dimensions. In this domain, a structured quadrilateral mesh... 

Finally, computational aspects and numerical issues are considered. This includes a short description of the finite difference, the finite volume, and the finite element discretization methods. The chapter ends with some general comments. 

The QC method which presents a relationship between the deformations of a continuum with that of its crystal lattice uses the classical Cau-chy-Bom rule and representative atoms. The quasi-continuum method mixes atomistic-continuum formulation and is based on a finite element discretization of a continuum mechanics variation principle. 

The soft soil is taken to consist of five layers, from a soft one at the surface to a compacted layer at the bottom and assumed to overlie a dense sandy soil that acts as the engineering bedrock where the ground acceleration is defined. The soil layer properties are described in Table 25.1. For simplicity, aU concrete slabs and the fotmdation mat are considered to have a uniform thickness of 0.5 m and are modelled as 3D elastic solid elements with unit weight 25 kN/m, elastic modulus 28,000 MPa and Poisson s ratio 0.2. The finite element discretization was carried out using 3D-nonlinear solid element for the soil and 3D-elastic solid element for RC slabs and the mat foundation, as depicted in Fig. 25.5. 

For instance, data from the vane shear test in clays are usually analyzed within a limit equilibrium solution, whereas pressuremeter results are considered within cylindrical cavity expansion. Alternative theoretical solutions proposed for analysis of CPT data include limit plasticity, strain path method, finite elements, discrete elements, hybrid cavity expansion-critical state, and dislocation theory. Usually, the approaches are established for two extreme cases of drainage, either (a) undrained, applied to clays or (b) fully-drained, applied to sands. In reality, many possible scenarios lie between the two conditions, as discussed by Randolph (2004) and Schneider et al. (2008). 

James and Williams (1985) employed a Galerkin finite elanent schane to provide flexible numerical solutions to the PB equation in one and two dimensions, using a Newton sequence for the solution of the set of nonlinear equations arising from the finite-element discretization procedure the procedure was shown to be applicable for different geometries. Dyshlovenko (2002) applied a finite-element solution of the PB equation with an adaptive mesh refinement procedure. The procedure allowed the gradual improvement of the solution and adjustment of the geometry of the problem. The approach was successfully applied to several problems... 

For the finite element discretization beam finite elements are used, with two degrees of freedom at each node the transversal deflection u>i and the rotation ijfi. They are gathered to form the degrees of freedom vector X = wi After assembling the mass and stiffness matrices for all elements, we obtain the equation of motion in the form ... 

This research project is funded by NSF through the Career Award CMS-9734473. The grain-based finite element discretizations have been obtained with the help of the mesh generation software TRIANGLE originally developed by J. R. Shewchuk. 

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