Mathematical modeling finite-element methods

The beauty of finite-element modelling is that it is very flexible. The system of interest may be continuous, as in a fluid, or it may comprise separate, discrete components, such as the pieces of metal in this example. The basic principle of finite-element modelling, to simulate the operation of a system by deriving equations only on a local scale, mimics the physical reality by which interactions within most systems are the result of a large number of localised interactions between adjacent elements. These interactions are often bi-directional, in that the behaviour of each element is also affected by the system of which it forms a part. The finite-element method is particularly powerful because with the appropriate choice of elements it is easy to accurately model complex interactions in very large systems because the physical behaviour of each element has a simple mathematical description. 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

The mathematical model of a MAT reactor, considering a 12 lump model, has been discretized using a finite element method in the direction of gas flow. The resulting system of differential-algebraic equations (DAEs) has been solved by an appropriate computer code (DASSL). 

Acar and coworkers (46] and Shapiro et al. [52] have presented general models based on the first of these two approaches. These models predict that the contaminant and the electrolysis products at inert electrodes will be transported and dispersed by advection, migration, and diffusion. Modelling in this manner provides only a first-order, mathematical framework to examine the flow patterns and chemistry generated in the process adsorption/desorption kinetics, acld/base chemical reactions, complex equilibria, and precipitatlon/solubility factors may heavily influence the model accuracy and outcome of any site remediation. Two approaches for mathematic modelling are the use of analytical solutions or numerical, finite element methods (FEM). Both models require adequate definitions for the boundary conditions (nature of electrolyses, flow behaviour). 

Dong P and Xu X. 1998. Mathematical Models for Fluid-Solid Coupling in Reservoir and its Finite Element Method Equations. ACTA Petrolesi Sinica, 1998,19(l) 64-70... 

Geometric tolerances are also used in design to limit the form deviations. Classical tolerance models are not yet satisfactory in simulating the effect of the form deviations on part geometry. The modal tolerancing method allows to decompose any real defects of any discretized feature in the natural mode shapes with an unambiguous mathematical language. The finite element method is used to compute those shapes, and... 

ABSTRACT In this paper, established the mathematical model of the process of thermal storage in the solar-ground concrete pile. It adopted finite element methods to numerically simulate the unsteady-state temperature field of concrete pile that around the underground vertical tube, given the calculated format of different boundary condition, analyzed the temperature variation rule of the concrete around heat exchanger. This paper provided reference basis of ascertain method about the bury depth of the vertical U-tube and the mix proportion of the concrete pile. 

Use the equivalent tube method established the physical and mathematical models for the temperature field surrounding of the vertical U-tube, and used the finite element method for solving the mathematical model. 

The present paper follows our two earlier contributions, Coimbra (2000 and 2002) where we have presented the formal treatment of moving finite element method with a piecewise higher degree polynomial basis in space. Without loss of generality we will only describe the MFEM in 2D. The mathematical model of a process involving diffusion, reaction and convections in 2D usually consists of an equation of the form... 

Coimbra, Maria do Carmo, Sereno, C. and Rodrigues, A.E., 2002, A moving finite element method for the solution of two-dimensional time-dependent models, Applied Numerical Mathematics, in press, available online 11 June 2002. 

Perre, P. and Passard, J., 1995. A control-volume procedure compared with the finite-element method for calculating stress and strain during wood drying. Drying Technol. J., Special Issue Mathematical Modelling and Numerical Techniques for the Solution of Drying Problems, 13(3) 635 60. 

Trials have been undertaken to evaluate the state of insulation using electrochemical techniques, in particular electrochemical impedance spectroscopy to determine the disbonding area, the break-point frequency method proposed by Haruyama et al. (1987), or the pseudocapacitance method . Field measurements require the formation of mathematical models of cathodical-ly polarized underground or underwater structures. The finite element method (FEM) and the boundary element method (BEM) are used, allowing prediction of the current and potential distribution of large structures covered with coatings, e.g., pipe-... 

Plate I (a) in the colour section between pages 130 and 131 shows a finite element model for a single representative unit of a scaffold. It is very easy to use the irregular tetrahedral elements to model devices of sophisticated shapes. The vertexes of the elements are named as nodes . The tetrahedral elements share a common set of concentrations at the nodes. The distribution of concentration inside an element is less important if the elements are sufficiently small. This is like controlling the shape of a fishing net by holding all the knots. Instead of trying to find a mathematical function to describe the concentration in the entire device, the finite element method finds discrete values of the concentration at the nodes. 

All local mathematical equations can be rewritten into a global principle, although many of the principles do not make an explicit sense such as minimising the potential energy. The finite element method can therefore be conveniently used to model intercoupling and multi-physics problems. 

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