Discrete element formulation

If the micromechanical behaviour and large deformations should also be taken into account, the Discrete Element formulation is predestined. 

Based on the pilot scale measurements mathematical modeling was started in order to describe the processes. Discrete element method (DEM) was used to model the mixing. This was originated from a simple mechanical model to formulate... 

Time delays can also be handled with the LQP, although the discrete-time formulation (46) of the LQP is better suited to the time delay problem (especially when there are only a few such elements in the differential equations). ... 

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. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

In order to obtain the boundary finite element formulation, on the one hand the force-displacement relation of the discretized element layer between nodal forces P and nodal displacements u is considered, which can be written in the decomposed form ... 

Using a finite element type formulation, the concentration of the discretized element Bt is... 

To build a pharmaceutical formulation expert system, the formulation process has to be broken down into a number of discrete elements in order to provide distinct problem-solving tasks, each of which can be reasoned about and manipulated. However, as the formulation process is so complex, none of these tasks can be treated independently. A means of representing interactions and communicating information between tasks is therefore required. For example, one task may result in certain preferences that must be taken into account by subsequent tasks. To achieve this level of communication between tasks, the information in an expert system has to be highly structured and is therefore often represented as a series of production rules. An example of a production rule is as follows ... 

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

W. A. Fiveland and J. P. Jessee, A Finite Element Formulation of the Discrete-Ordinate Method For Multidimensional Geometries, in Radiative Heat Transfer Current Research, ASME HTD no. 244, New York, 1993. 

The model is solved numerically by means of the finite element code ASTER. The equations are discretized within a finite element formulation. The time discretization is implicit and the coupling is solved by means of a global inversion of the system, as explained in Chavant (2(X)1). 

Finite element formulation involves subdivision of the body to be modeled into small discrete elements (called finite elements). The system of equations represented from 4.48 to 4.59 are solved for at the nodes of these elements and the values of mechanical displacements u and forces F as well as the electrical potential d> and charge Q. The values of these mechanical and electrical quantities at an arbitrary position on the element are given by a linear combination of polynomial interpolation functions N(x, y, z) and the nodal point values of these quantities as coefficients. For an element with n nodes (nodal coordinates (x y z) f = 1, 2,..., n) the continuous displacement function m(x, y, z) (vector of order three), for example, can be evaluated from its discrete nodal point vectors as follows (the quantities with the sign are the nodal point values of one element) ... 

To simulate transient thermal behavior of the battery, a heat balance is formulated for each discretization element ... 

Section 2.1.3 shows how the process of diffusion in one dimension can be represented by a chain of resistors and capacitors, and Section 2.1.6 shows how porous electrodes can be represented by a similar network. While this approach is valid even for a distributed process with no boundary (like diffusion into infinite space), discretization is even more important for the case where a distributed process is limited in space. In this case, a finite number of discrete elements can describe the system to arbitrary precision, and can be used for numerical calculations, as treated in next chapter, even if no analytical solution is possible. Another convenience of discretizing a distributed process is the resulting ability to add additional subprocesses directly to the equivalent circuit rather than starting the derivation by formulating a new differential equation. For example the equivalent network representing electric response of a pore is given in Figure 4.5.3. 

To describe the finite element formulation of the discretized Hele-Shaw equation, we write the Hele-Shaw equation in the following form... 

The cracked concrete is anisotropic and these relations must be transformed to the reference axes XY. The simplified averaging process is more convenient for finite element formulation than the singular discrete model. A smeared representation for the cracked concrete implies that cracks are not discrete but distributed across the region of the finite element. The smeared crack model used in this work is based on the assumption that the field of multiple micro-cracks is created (Cervenka, 1985). The validity of this assumption is determined by the size of the finite element, hence its characteristic dimension = 4a, where A is the element area (versus integrated point area of the element). 

Lemos JV (2007) Discrete element modeling of masonry structures. Int J Archit Herit 1(2) 190-213 Lourenfo PB (1996) A matrix formulation for the elastoplastic homogenisation of layered materials. Mech Cohes-Frict Mater 1 273-294 Lourenfo PB (2000) Anisotropic sohening model for masonry plates and shells. J Struct Eng 126(9) 1008-1016... 

This is Navier s equation of elastodynamics. Using the standard Galerkin method, one can obtain the weak form of this equation and then discretize the problem in space. This procedure entails the introduction of set of arbitrary functions 0, known as the test fimctions. The test functions are auxiliary fimctions which help formulate an approximate solution u to the displacements u, called the trial functions. The domain Q is then discretized in space using a set of global piecewise linear basis functions 4>, which divide the domain into discrete elements Q. As a result, both the test and trial functions become linear combinations of the global basis functions,... 

In the basis set formulation, we need to evaluate matrix elements over the G-H basis functions. We can avoid this by introducing a discrete variable representation method. We can obtain the DVR expressions by expanding the time-dependent amplitudes a (t) in the following manner ... 

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... 

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

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... 

These concepts were implemented according to the following scheme the liquid element surrounding the bubble and the bulk are considered as two separate dynamic reactors that operate independent of each other and interact at discrete time intervals. In the beginning of the contact time, the interface is being detached from the bulk. When overcome by the bubble, it returns to the bulk and is mixed with it. Hostomsky and Jones (1995) first used such a framework for crystal precipitation in a flat interface stirred cell. To formulate it for a... 

In tlie case of a discrete sample space (i.e., a sample space consisting of a finite number or countable infinitude of elements), tliese postulates require tliat tlie numbers assigned as probabilities to tlie elements of S be noimegative and have a sum equal to 1. These requirements do not result in complete specification of tlie numbers assigned as probabilities. The desired interpretation of probability must also be considered, as indicated in Section 19.2. The matliematical properties of the probability of any event are tlie same regardless of how tliis probability is interpreted. These properties are formulated in tlieorems logically deduced from tlie postulates above without tlie need for appeal to interpretation. Tliree basic tlieorems are ... 

---