2.4 Boundary Conditions
3.1.5 Shape Functions
The global shape functions are defined as: ϕi(xj) =
1 if i = j , 0 if i6= j ,
where i, j are nodes within a given mesh. The global functions are composed of element shape functions, ψ, which are defined on each element that supports a specific global function.
The shape functions can be taken from various members of the set of mathematical functions. Some popular choices are Bessel, exponential, trigonometric, or polynomial functions. The use of polynomials can facilitate faster frequency sweeps; more will be said of this in Section 3.2.2. As such, only polynomial shape functions will be considered in this work. There are many different kinds of polynomial functions to choose from, like for example Chebyshev, Lagrange, or Legendre polynomials.
As an example, we will briefly introduce the linear triangular Lagrange shape func- tions. There are three shape functions associated with the linear triangular functions,
3. FINITE ELEMENT METHODS FOR WAVE PROBLEMS 0 1 0 10 1 η ζ ψ1 0 1 0 1 0 1 η ζ ψ2 0 1 0 1 0 1 η ζ ψ3
Figure 3.3: The three shape functions associated with linear triangular elements.
and these are given by:
ψ1= 1− η − ζ , ψ2 = η , and ψ3= ζ . (3.2)
These functions are defined on the reference element shown in Figure 3.2. The functions are depicted in Figure 3.3. The shape functions are unity at one vertex, and have zero value at the remaining vertices. This property ensures interelement continuity.
A discussion of the key properties of shape functions follows.
3.1.5.1 Continuity
Let us consider the continuity requirements of the problem to be modelled. To satisfy the weak formulation of the problem, the shape functions must be CG−1 continuous between the elements, and CGdifferentiable inside the elements, where G is the highest
order derivative in the weak formulation [80].
For a second order differential equation, like the convected wave equation, the func- tions used must be C0 continuous between the elements. This implies that the solution field is constructed using a continuous function whose first derivative is discontinuous at element interfaces. Some structural mechanics models require C1 continuity, in that case the shape function and its first derivative must be continuous, and its second derivative may be discontinuous between the elements.
Most of the methods considered in this work are equipped with C0 continuity, but
note that we will also consider the Hermite method (to be introduced in Section 3.5) which ensures C1 continuity.
3.1 The Standard Method 0 0.5 1 0 0.5 1 η ψ
Figure 3.4: Modal shape functions on a one-dimensional element with two nodes. The
shape function which is non-zero on the left node is given by the dashed blue line, and the function which is non-zero on the right node is shown using the dashed red line. Four bubble functions are shown.
3.1.5.2 Higher Order Modal Expansion
The accuracy of the finite element solution can be improved either by increasing the number of elements used, this is referred to as h-refinement, or by increasing the poly- nomial order of the shape functions, which is referred to as P -refinement. Increasing the polynomial order of the shape functions can be carried out in two ways. One method is to increase the number of nodes in the elements, higher order shape functions are then introduced using these nodes. These kinds of shape functions are called nodal, and an example is the high order Lagrange functions. The other method is to use a modal expansion of the nodes which are already available; such functions are called modal. The shape functions which will be investigated in this work are of the modal type.
In one dimension , as the interpolation order P is increased, i.e. for P ≥ 2, bubble functions are added to the approximation space. Bubble functions are zero at both nodes, but have non-zero amplitudes inside the element. This property of the shape functions is demonstrated in Figure 3.4, which shows the one-dimensional modal Bern- stein shape functions. In this case, the polynomial order of the shape functions is 5, and thus two nodal functions and four bubble functions are found. A higher order shape
3. FINITE ELEMENT METHODS FOR WAVE PROBLEMS
x y
z
Figure 3.5: Degrees of freedom of a four-noded tetrahedral element with modal shape
functions. Node functions, edge functions•, face functions ◦.
function, given in terms of a modal expansion, can be written as:
ψ(η) = a1ψ1(η) + P
X
i=2
aiψi(η) + aP +1ψP +1(η) ,
where an is the coefficient multiplied by function ψn, and ψ1 = (1− η)/2 and ψP +1 =
(1 + η)/2 are the linear nodal functions attached to the first and last node, respectively. ψi are bubble functions, where ψi(η) = 0 for η =±1.
In two dimensions, node, edge and face (or 2D bubble) functions are found. For three-dimensional elements, node, edge, face and volume (3D bubble) functions are found. As an example, consider Figure 3.5, which depicts the degrees of freedom different types of functions on a tetrahedral element with four nodes. Node functions are zero on all nodes but one, edge functions are zero on all other edges, face functions are zero on all other faces, and bubble functions vary only within the element. As will be seen in Section 3.1.5.4, bubble functions can be removed from the final system to be solved without loss of accuracy, but with the added benefit of improved conditioning (e.g. [97]).
3.1 The Standard Method
3.1.5.3 Hierarchic Functions
A disadvantage of using high order nodal functions is the need to construct a new set of shape functions when the polynomial order is increased. This has implications for P -refinement; increasing the order will require the construction of a new mesh. If instead modal shape functions are used, the polynomial order can be increased without re-meshing. Although this is clearly an improvement in terms of efficiency, it still requires the construction of new element matrices when the order is changed. If modal shape functions which do not change with the order are used instead, further savings can be made; as higher order functions can be included in the existing approximation space without changing lower order shape functions. Such shape functions are called hierarchic.
This property is ideal for P -adaptive schemes, in which case a local estimator could be used to determine the optimum interpolation order required in different regions of a problem, and the necessary shape functions sets could then be appended to the element matrices. All of the functions, up to the highest order required, could then be stored and accessed as needed by the global system assembly procedure. This property enables a very efficient implementation of the finite element method. In this work, the Lobatto method (to be introduced in Section 3.6) makes use hierarchic functions.
3.1.5.4 Static Condensation
As the polynomial order of the shape functions is increased, the number of bubble functions increases. Bubble functions only contribute to the solution locally - they have no connectivity with neighbouring elements. The bubble functions can be removed from the final system using a technique called static condensation, which involves removal of the internal degrees of freedom from the set of global degrees of freedom, at the element level. Making use of this technique means that only the degrees of freedom which are shared between elements are included in the final global matrix. The solution values interior to the elements can be recovered during post-processing.
Using condensation has two advantages. Firstly, it reduces the size of the final sys- tem, thus reducing the computational cost of solving the system. Secondly, it improves the conditioning of the system, particularly for higher order polynomial approximations
3. FINITE ELEMENT METHODS FOR WAVE PROBLEMS
[137]. It is worth noting that this process does not affect the final solution. Condensa- tion is performed by decomposing the element matrix and the unknown solutions into exterior and interior components, as follows:
"Aee Aei Aie Aii # ae ai ="Bee 0 0 0 # ae ai
where (·)e are the external degrees of freedom, and (·)i are the internal degrees of
freedom. This can be simplified to give Bee= Aee−AeiA−1ii Aie, the condensed matrix,
where Aee− AeiA−1ii Aie is the Schur complement. The final system to be solved is
then Beeae = f . A more rigorous description of the condensation technique can be
found in Reference [150].