6.3 Comparison of different one-dimensional Ansatz functions
6.3.1 Hierarchic vs. non-hierarchic shape functions
One important advantage of hierarchic over non-hierarchic sets of basis functions is indi-cated in Fig. 6.6.
K F
p= 1 p= 2 p= 3
Figure 6.6: Hierarchic structure of stiffness matrix and load vector for p= 3.
Shape Functions 88 When increasing the polynomial degree by one order, only one shape function needs to be added to a hierarchic set [115], while a non-hierarchic set of Ansatz functions has to be generated anew (one-dimensional case). If an element refinement is necessary, the totally new set of shape functions requires all calculations to obtain the system matrices to be re-peated [241]. This feature has an immediate effect on the structure of the system matrices.
That is to say, the system matrices and the load vector corresponding to the polynomial order pare a sub-set of the global system matrices and the load vector for the polynomial degreep+ 1. In Fig. 6.6, we use different shades of grey to illustrate the advantage of hier-archic Ansatz functions concerning p-refinement procedures. When the polynomial degree for the approximation of the unknown field is successively increased, only those parts of the system matrices that are indicated in brighter shades need to be computed while all components of the matrices have to be re-calculated in case of a non-hierarchic set of shape functions.
Another point we have to take into account is that the degrees-of-freedom have no physi-cal interpretation in the p-version of the FEM [215]. If Lagrange-type shape functions are employed the degrees-of-freedom correspond to the nodal values of the independent field variables. In the case of hierarchic Ansatz functions, only the nodal modes can be inter-preted as field variables, whereas all higher order degrees-of-freedom are merely unknown parameters determining the solution space. A more involved post processing is therefore required when dealing with hierarchic Ansatz functions.
There are further advantages connected to the hierarchic structure of the set of shape functions and the fact that it constitutes a modal basis instead of a nodal one, such as the p-version FEs’ inherent ability for a local p-refinement. Due to the fact that we can freely choose the polynomial degree for all edges, faces and the interior of the p-element, a local, adaptivep-refinement seems easily feasible. These special transition elements can be derived automatically without any additional considerations. We only have to ensure the compatibility at adjacent faces and edges between neighboring FEs [117, 119], cf. Section 6.4. Moreover, the hierarchic nature of the shape functions itself facilitates an easy imple-mentation of adaptive refinement strategies (structure of the system matrices). In contrast to the behavior of p-elements, a local refinement cannot be easily achieved by using both SEs and conventional FEs. Both schemes need especially generated transition elements to ensure continuity. These special kinds of FEs are, however, difficult to develop.
One of the main reasons why the p-version of FEM is hardly used for ultrasonic guided wave propagation analysis is that there is no mass-lumping technique available yet. How-ever, a diagonal mass matrix is indispensable in order to fully exploit the advantages of an explicit time integration. If we use a consistent mass matrix formulation, a system of equations has to be solved for each time-step - while a diagonalized mass matrix re-quires only matrix-vector-operations. It is therefore highly advantageous to be able to employ explicit time-marching schemes in wave propagation analysis. Remember, that the methods developed in the present thesis are primarily applied to ultrasonic guided wave propagation problems. In consequence, it is imperative to have efficient time-integration methods at hand. As the time-step is naturally limited by the high-frequency regime, ex-plicit time-stepping algorithms are only advantageous over imex-plicit ones if the mass matrix is a diagonal matrix [151, 152]. In this thesis, we employ an implicit algorithm such as the Newmark-method (cf. Section 4.3.3) where a consistent mass matrix formulation is cho-sen. Yet, the use of implicit time-marching schemes for high frequency wave propagation problems tends to be numerically costly [78]. Considering FEMs based on Lagrange-type
Shape Functions 89 shape functions, we can diagonalize the mass matrix by using a special nodal quadrature rule, the row-sum technique or the HRZ-lumping method [151–153] (cf. Section 4.6), to ensure that explicit time integration schemes become feasible, cf. Section 4.3.2.
Trigonometric functions possess some advantages over polynomial Ansatz functions. They are indefinitely derivable and no recurrence formulae are needed to generate them, which serves to minimize the round-off errors [279, 280, 283]. These two features result in a bet-ter conditioned system of equations compared to other kinds of hierarchic shape functions, cf. Section 6.3.2. To date, these shape functions are almost exclusively used for modal analysis applications.
Table 6.1: Comparison of key features of the different shape function types (one-dimensional case).
hhhhhhh
hhhhhhhhhhhhh
Feature
Method
FEM SEM p-FEM Fourier-p-FEM
Inter-element continuity C0 C0 C0 C0
Degrees-of-freedom (interpretation) physical physical unknowns unknowns
Mass-lumping yes yes no no
row-sum row-sum -
-nodal- nodal- -
-quad. quad.
HRZ HRZ -
-Hierarchic set of functions no no yes yes
Number of common
degrees-of-freedom between adjacent elements 1 1 1 1
Runge phenomenon (oscillations) yes no no no
One frequently mentioned drawback of higher order Ansatz functions based on an equidis-tant nodal distribution is the occurrence of the so-called Runge phenomenon [87, 183], cf.
Fig. 6.7. This effect is related to the interpolation of functions. According to [290–293], the idea to interpolate functions deploying a polynomial interpolation scheme in equis-paced points is fundamentally wrong. In case we want to interpolate a smooth function by polynomials in p+ 1 equally spaced points, the error may increase at a rate of 2p and therefore the approximation fails to converge even if p→ ∞ [291]. Here, oscillations with high amplitudes close to the boundaries of the interval are a characteristic sign, because this is where the error between the original function and its interpolation polynomial in-creases rapidly [87], cf. Fig. 6.7. As mentioned before, this can be observed when applying equidistant nodal distributions, rendering them unsuitable for the interpolation of higher order polynomial functions. We can minimize these oscillations, however, by using nodal distributions that are denser (clustered) at the ends of the interval (unevenly spaced points) [292]. Such distributions decrease the Lebesgue constant [293]. The Lebesque constant is a measure of how suitable the interpolant of a function (at the chosen nodal distribution) is in comparison with the best polynomial approximation of the function (it therefore bounds the interpolation error).
Shape Functions 90
-0.4-0.20.20.40.60.81.21.401
-0.8 -0.4 0 0.4 0.8
Functionvalue
ξ
(a)Equispaced nodal distribution: N= 6
-0.4-0.20.20.40.60.81.21.401
-0.8 -0.4 0 0.4 0.8
Functionvalue
ξ
(b)GLL nodal distribution: N = 6
-0.4-0.20.20.40.60.81.21.401
-0.8 -0.4 0 0.4 0.8
Functionvalue
ξ
(c)Equispaced nodal distribution: N= 16
-0.4-0.20.20.40.60.81.21.401
-0.8 -0.4 0 0.4 0.8
Functionvalue
ξ
(d) GLL nodal distribution: N = 16
-0.4-0.20.20.40.60.81.21.401
-0.8 -0.4 0 0.4 0.8
Functionvalue
ξ
(e) Equispaced nodal distribution: N = 31
-0.4-0.20.20.40.60.81.21.401
-0.8 -0.4 0 0.4 0.8
Functionvalue
ξ
(f) GLL nodal distribution: N = 31 Annotations: The function to be interpolated is the witch
of Agnesi curve -f(ξ) = 1/ 1 + 16ξ2
- a classical example to demonstrate that the interpolation error can increase without bounds.
Witch of Agnesi curve
Lagrange interpolation polynom.
Nodal distribution
Witch of Agnesi curve Lagrange interpolation polynom.
Nodal distribution
Figure 6.7: Runge phenomenon: Comparison of Lagrange interpolation polynomials through an equidistant nodal distribution and a GLL one.
Shape Functions 91 For a detailed investigation of polynomial interpolation in the context of higher order methods, the interested reader is referred to [290–293] and the references cited therein.
Tab. 6.3.1 summarizes selected properties of the discussed one-dimensional shape functions. Thus, their advantages and disadvantages can be judged at a glance.