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N. Bhardwaj et al.
A brief introduction to FEM is given in the next section followed by its implementation into FSW modeling for thermal, material flow and mechanical properties
analysis. An introduction to cellular automata finite element (CAFE) is also given,
which is gaining popularity for predicting microstructure evolution during FSW.
3.5.1 Basics of Finite Element Modeling
A method used to numerically solve differential as well as integral equations by
discretizing a body into small finite elements is referred to as finite element method
(FEM). The method is useful for numerically solving both ordinary and partial differential equations. The solution is arrived at by assuming a piecewise continuous function and obtaining parameters of the function such that error in the solution reduces.
It can be applied to solve physical problems when the governing equations of the
physical phenomenon are available. The steps involved in FEM are as follows: (i)
pre-processing that involves mesh generation, (ii) obtaining the assembled system
of equations, for which the elemental matrices and vectors are to be evaluated, (iii)
determining the boundary conditions and applying them, (iv) solving the system
of equations and (v) post-processing. Finite difference approximation is taken for
solving time-dependent problems by treating the derivative with time [25].
In FEM, the continuum body is discretized into subdivisions called finite elements
that are connected to each other via joints called nodes/nodal points. The field variables (e.g., displacement, temperature, stress, etc.) are approximated using a simple
function (approximating function or interpolation models). The approximating functions are defined using the nodal field variables. A polynomial form is considered
for the solution or interpolation model. The new unknowns in the field equations for
the whole continuum will be values of the field variables at the nodes. After solving
the set of finite element equations, the nodal field variables will be known which
will lead to approximation of the field variables in the whole continuum. Since a
complicated physical problem is converted to a simpler problem, only an approximate solution is obtained. However, the approximate solution can be refined and
improved by reducing the error by using some computational efforts. The method is
explained with the following simple example.
A linear differential equation can be represented as
L u + q = 0,
(3.75)
where u is the vector of primary variables of the problem as a function of the coordinates, q is the vector of known functions, and L represents a differential operator. The
differential equation is subjected to natural and essential boundary conditions. The
conditions that are sufficient for completely solving a differential equation are called
essential boundary conditions. However, the natural boundary conditions contain
higher-order derivatives and cannot solve the differential equation completely. For
example, considering the differential equation
N. Bhardwaj et al.
A brief introduction to FEM is given in the next section followed by its implementation into FSW modeling for thermal, material flow and mechanical properties
analysis. An introduction to cellular automata finite element (CAFE) is also given,
which is gaining popularity for predicting microstructure evolution during FSW.
3.5.1 Basics of Finite Element Modeling
A method used to numerically solve differential as well as integral equations by
discretizing a body into small finite elements is referred to as finite element method
(FEM). The method is useful for numerically solving both ordinary and partial differential equations. The solution is arrived at by assuming a piecewise continuous function and obtaining parameters of the function such that error in the solution reduces.
It can be applied to solve physical problems when the governing equations of the
physical phenomenon are available. The steps involved in FEM are as follows: (i)
pre-processing that involves mesh generation, (ii) obtaining the assembled system
of equations, for which the elemental matrices and vectors are to be evaluated, (iii)
determining the boundary conditions and applying them, (iv) solving the system
of equations and (v) post-processing. Finite difference approximation is taken for
solving time-dependent problems by treating the derivative with time [25].
In FEM, the continuum body is discretized into subdivisions called finite elements
that are connected to each other via joints called nodes/nodal points. The field variables (e.g., displacement, temperature, stress, etc.) are approximated using a simple
function (approximating function or interpolation models). The approximating functions are defined using the nodal field variables. A polynomial form is considered
for the solution or interpolation model. The new unknowns in the field equations for
the whole continuum will be values of the field variables at the nodes. After solving
the set of finite element equations, the nodal field variables will be known which
will lead to approximation of the field variables in the whole continuum. Since a
complicated physical problem is converted to a simpler problem, only an approximate solution is obtained. However, the approximate solution can be refined and
improved by reducing the error by using some computational efforts. The method is
explained with the following simple example.
A linear differential equation can be represented as
L u + q = 0,
(3.75)
where u is the vector of primary variables of the problem as a function of the coordinates, q is the vector of known functions, and L represents a differential operator. The
differential equation is subjected to natural and essential boundary conditions. The
conditions that are sufficient for completely solving a differential equation are called
essential boundary conditions. However, the natural boundary conditions contain
higher-order derivatives and cannot solve the differential equation completely. For
example, considering the differential equation