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Subsections
5.2 Discretization with Tetrahedrons
An often used choice is to discretize the solution domain with tetrahedrons. On the one side a tetrahedron is a relative simple element, especially regarding meshing aspects, on the other side it is an efficient element to discretize structures with nonplanar surfaces or complex geometries. As shown in Fig. 5.1, this element is limited by four triangles and has four vertexes which are in any case grid nodes in the mesh.
For using the weighted residual method the shape functions must be continuous on the transition from one element to its neighbor element. Within the elements they must be at least onetime differentiable. The shape functions will be defined locally on the tetrahedron. It should be noted that the global shape function
is assembled from the local shape functions of the elements which share the same node .
If it is assumed that the discretization is carried out with linear shape functions, the four vertexes used are the four grid nodes on the element. This means that the shape functions must depend on the x, y, and zcoordinate linearly, so that in general form it can be expressed by [90]

(5.8) 
are the numbers of the local grid node. Since for every grid node a separate shape function is needed, there are four shape functions on a four node element. The coefficients , , , and must be determined in such a manner, that the respective shape function fulfills (5.4). This means, that for example the value of the shape function (x,y,z) must be 1 in node P with its coordinates (,,), and 0 in all other nodes. With this information the following equation system with N on the four nodes can be written:
With this equation system it is possible to determine the four unknown coefficients , , , and for the shape function (x,y,z).
Figure 5.1:
Tetrahedral element in a global coordinate system

Figure 5.2:
Tetrahedral element in a normalized
coordinate system

For calculation of the unknown coefficients Cramer's rule can be applied, which says: if there is an equation system
, the numbers

(5.10) 
are the components of the solution .
With Cramer's rule, for example, the coefficient from the shape function
can be calculated by

(5.11) 
where the matrix
is

(5.12) 
By deleting the row i and the column j of a nrowed determinant, a new (n1)rowed subdeterminant
with sign
is constructed

(5.13) 
To simplify the arithmetic, a nrowed determinant can be calculated with the sum of n (n1)rowed subdeterminants (5.13). The determinant can be expanded from a row or column. If for example is the number of any column, than the determinant is

(5.14) 
With this rule (5.14) the coefficient (5.11) can be simplified, because only
, and the rest in the first column is 0. After expanding the determinant from the first row one obtains

(5.15) 
In the same way all other coefficients of the shape function can be determined

(5.16) 
The coefficients for the other shape functions can be found in the same way, the only difference leis in the equation system (5.9). Because, for example,
must be 1 in node P and 0 in all other nodes, it can be formulated
With the above described procedure also the coefficients for
can be determined straightforwardly to

(5.18) 
5.2.2 Coordinate Transformation
A coordinate transformation can help to simplify the calculation of integrals. For constructing the residual the calculation of the following element integral is frequently needed

(5.19) 
Here the multiplication of two (linear) form functions leads to a more complex polynomial which complicates the integration over the region. It is more practical to integrate over a normalized element (see Fig. 5.2). For this purpose, a tetrahedron with any location in the global (x,y,z)coordinate system must be transformed into a normalized local
coordinate system.
Each point (x,y,z) of the tetrahedral element in the global coordinate system can be transformed to a corresponding point
in the normalized coordinate system with the following bijective projection rule
This projection in matrix form leads to

(5.21) 
and the conversion from the global to the normalized coordinates is

(5.22) 
with
and
.
is the socalled Jacobian matrix which only depends on the global coordinates (x,y,z)

(5.23) 
The element integral (5.19) calculated in the normalized coordinate system must be multiplied with the determinant of the Jacobian matrix

(5.24) 
because the following relationship holds

(5.25) 
The shape functions for the normalized tetrahedron are simpler than those in the global coordinates, because they are reduced to [91]
These shape functions lead to a simpler integrand in (5.24). A further advantage is that after normalization the lower integration limit is always 0, because as shown in Fig. 5.2, the tetrahedron starts in the origin of ordinates.
Also the upper limits can be found straightforwardly. As shown in Fig. 5.2, the element
is bounded by a plane which goes through the points P(1,0,0), P(0,1,0), and P(0,0,1). This plane is described with the equation
.
The maximum on the axes is 1. The limit in the plane () can be described with
and the limit in direction is
.
With these limits the element integral of can be written in the form

(5.27) 
This is a real advantage of the normalized tetrahedron, because there exists a simple scheme for . The result from integration over the element is either
or
, only depending, if the two functions and are equal or not. So in fact, to calculate the integral in the normalized coordinate system, only the determinant of the Jacobian matrix must be calculated. This procedure is much easier than to find the element integral for a common tetrahedron in the global coordinate system, where each element has a different size. This means that each integral has a different result and must be calculated separately.
The differentiation of the whole projection (5.2.2), with respect to leads to
with respect to leads to
and with respect to leads to
These derivatives can be also expressed in matrix form

(5.31) 
and the following relationship for the partial differential operators in the normalized system can be found with

(5.32) 
The inverse of the Jacobian matrix (5.23) in the global coordinate system is given by

(5.33) 
where the components of the inverse matrix are
The components of the Jacobian matrix (5.23) depend only on the location of the tetrahedron vertices in the global coordinate system. Because of

(5.35) 
there exists a relationship between the partial differential operators in the normalized system and the coordinates of the four nodes from the global system.
If any continuous function
in the normalized coordinate system is differentiated in respect to , then the chain rule must be used so that

(5.36) 
The gradient of the function
in the normalized system becomes to

(5.37) 
This means that the gradient operator in the normalized system must be multiplied with the transposed of the inverse Jacobian matrix

(5.38) 
5.2.4 Discretization of the Oxidant Diffusion
In the continuum formulation from (3.2), the diffusion of oxidants through the oxide material is described with

(5.39) 
When Galerkin's method is applied, it is multiplied with a weight function
and integrated over the domain , which leads to

(5.40) 
If there are two functions
and
defined on a domain , Green's theorem says that

(5.41) 
where is the boundary of the domain.
With Green's theorem the Galerkin formulation from (5.40) can be rewritten in the form

(5.42) 
Here the diffusion coefficient and the maximal reaction rate do not directly depend on the location and, therefore, they do not need to be integrated over space and can stand outside of the integral. Furthermore, it is assumed that there is no flow of oxidants through the boundary surface and the boundary condition becomes

(5.43) 
With this Neumann boundary condition the Galerkin formulation for the oxidant diffusion can be reduced to

(5.44) 
With the finite element method it can be assumed that this equation is only valid on a single tetrahedral element . Furthermore, the scalar function for the oxidant concentration
is here approximated linearly with

(5.45) 
where
is the value of the oxidant concentration in node and at discrete time .
is the respective shape function from this node.
The distribution of the normalized silicon
is approximated in the same way so that

(5.46) 
where
is the value of the normalized silicon in node and at discrete time .
is the linear shape function (5.8) for node .
With the approximation for
and
, the oxidant diffusion on a single element can be described with

(5.47) 
In the approximation for
and
the shape function
is the same. Since the values and in the nodes do not depend on the spatial location, (5.47) can be rewritten as

(5.48) 
where it is assumed that
.
By substituting the integrals with
the discretized equation for the oxidant diffusion is simplified to

(5.50) 
The components were already calculated in Section 5.2.2. After integration of
in the normalized coordinate system, it was found that

(5.51) 
For calculating the components the integral is also transformed from the global to the normalized coordinate system, and with (5.25) follows

(5.52) 
where
are the shape functions for a normalized tetrahedron from (5.26).
It was demonstrated in (5.38) that the transformation of the gradient operator is carried out by a multiplication with the matrix
, so that the integrand from (5.52) becomes
After the multiplication of the two vectors and rearranging of this scalar product, (5.2.4) is
Here the terms which only depend on the location of the nodes in the global coordinate system, can be replaced by six constant coefficients
With (5.55) the scalar product (5.54) can be simplified to
For example, the simplified scalar product (5.56) for ,
is

(5.57) 
and for , it is

(5.58) 
For all combinations of
the simplified scalar product can be written in the form

(5.59) 
Instead of finding the scalar product (5.56), and the components of the matrices
for all combinations of
by the way like in (5.57), it is more comfortable to use

(5.60) 
and get the matrices
with

(5.61) 

(5.62) 
Because the derivatives of the linear shape functions
can only result in the values , 0 or , the integral from (5.52) becomes

(5.63) 
which means that all matrices
must be weighted with
.
After finding
and integration over the (normalized) element, the coefficients are

(5.64) 
Galerkin's method assumes that the residual from the (discretized) equation of the oxidant diffusion is zero, and so (5.50) is rewritten in the form

(5.65) 
This is a system with four equations, but with eight unknown variables
and
. Because of more unknowns than equations it is impossible to solve this equation system in present form. In the next section the required 4 equations introduced.
The dynamics of the normalized silicon concentration in the continuum formulation (3.5) is described by

(5.66) 
After applying Galerkin's method with a weight function
one obtains in a domain is

(5.67) 
Because of the time dependence in this equation an additional time discretization of the term
is necessary. This time discretization is performed with a simple backwardEuler method as

(5.68) 
where and are two successive discrete times.
For an equation
two Euler methods can be applied. The forwardEuler method is an explicit simple method, because the new value
is defined in terms of values that are already known [92].
The backwardEuler method comes from using at the end of a time step, when . It is an implicit method, because in order to obtain the new discrete value
a linear equation of the form
must be solved [92], which requires additional computing time. But compared with the forwardEuler method the most important advantage of the backwardEuler method is that a much larger time step size can be used. The reason is that implicit methods are usually much more stable for solving a stiff equation. A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the time step size is taken to be extremely small [93].
The spatial approximation for the oxdiant concentration
and the normalized silicon
for one finite element is the same as in Section 5.2.4
With the time discretization (5.68) and the spatial approximation for
and
, the Galerkin formulation for the dynamics on a finite element becomes

(5.71) 
Because of the same shape function for
and
and no spatial dependence of
and
, the last equation can be rearranged to

(5.72) 
After substituting
from (5.51) the discretized equation for the dynamics is simplified to

(5.73) 
In order to fulfill Galerkin's demand that the residual should be zero, the last equation can be rewritten as

(5.74) 
which is also a system with four equations and eight unknown variables
and
. The values for
are already determined at the previous time step.
By combining the two equation systems (5.65) and (5.74), a nonlinear but fully determined equation
system for one finite element, with 8 equations and the 8 unknowns
and
, is obtained. The system is nonlinear because of the product
in (5.65) and in (5.74). The complete equation system can be solved (for example with the Newton method) at each time point and the values for
and
can be determined.
The main interest in the continuum mechanics is the deformation of a body by internal or external forces. The deformation is expressed by the displacements . The displacement of a point in a threedimensional elastic continuum is defined by three displacement components , , and in directions of the three coordinates , , and , so that

(5.75) 
In contrast to the previous differential equations (5.39) and (5.66) for the mechanics Galerkin's method is not needed. Instead the virtual work concept is used [94]. The displacement components , , and are directly discretized on a finite tetrahedral element
where , , and are the displacement values in , , and direction on node
and is the linear shape function from (5.8).
The strain components in the elastic case are first order derivatives of the displacement components

(5.77) 
The element displacement is defined by the 12 displacement components of the 4 nodes as

(5.78) 
The submatrix
of displacement derivatives for the node is [95]

(5.79) 
The coefficients , , and are the same as already presented in (5.16) and (5.18).
The entire inner virtual work on a continuous elastic body, and so also on a finite element is [94]

(5.80) 
with the stress tensor
(3.15). Here it is assumed that there is no residual stress
, and the stress tensor is

(5.81) 
In discretized form the transposed strain tensor is

(5.82) 
After discretization the stress tensor (5.81) can be arranged as a function of the element displacement vector

(5.83) 
Together with the transposed strain tensor (5.82), this stress tensor leads to the following discretized form of the equation for the inner virtual work on a finite element

(5.84) 
The outer virtual work on a finite element, caused by the external nodal forces is

(5.85) 
because it is assumed that during the oxidation process there are not external forces acting.
On any elastic body, and so also on a finite element, the inner work must be equal with the outer work

(5.86) 
which can be simplified to

(5.87) 
Here the integrals can be substituted as sketched in [95]
where
is the socalled stiffness matrix and
can be declared as internal force vector. Since the integrands are not functions of the , , or coordinates, the integration over the volume is equal with its much more simpler multiplication. The volume of any tetrahedron in the global coordinate system can be calculated with the determinant of matrix (5.12) by

(5.90) 
The most important fact is that the residual strain tensor
in (5.89) loads the mechanical system. Because the residual strain components
are directly proportional to the normalized additional volume (3.37), there is a relationship between the volume expansion and the internal nodal forces.
With the integral substitutions (5.88) and (5.89), the balance equation (5.87) becomes a linear equation system for the mechanical problem on one finite element

(5.91) 
The system is fully determined, because there are 12 equations and also 12 unknown displacementcomponents (three on each node) on the tetrahedron.
Next: 5.3 Assembling and Solving
Up: 5. Discretization with the
Previous: 5.1 Basics
Ch. Hollauer: Modeling of Thermal Oxidation and Stress Effects