16.1 INTRODUCTION
Finite element method (FEM) provides an approximate solution of differential equations. There is a need to calculate the error, which is the difference between the exact solution and the approximate solution (i.e, FEM). Error in finite element solutions are divided mainly into three categories:
1. Domain approximation error, which is due to the approximation of the domain.
2. Quadrature and finite arithmetic errors, which are due to the numerical evaluation of integrals and the numerical computation on a computer.
3. Approximation error, which is due to the approximation of the solution.
In the formulation of the finite element method, usually the displacement or primary variable field is approximated by polynomials. This approximation is the main source of error in the solution. As this error is inherent in the method, the amount of error in the solution must be determined in order to judge the quality of the results obtained. This error information can be used to improve the results, by improving the primary variable approximation, and to monitor the convergence of the solution. The differences between the exact and approximate solution, i.e. errors decrease as the size of the subdivision ‘h’ gets smaller or as ‘p’, the order of the polynomial in the trial function used, increases.
In this chapter, various error measures are described. Two common estimates of error are a
priori and a posteriori error estimates. A discussion regarding these is presented. Recovery
method of estimating error is discussed in detail.
16.2 ERROR MEASURES
The subject of error estimation for finite element solutions and a consequent adaptive analysis, in which the approximation is successively refined to reach predetermined standards of accuracy, is central to the effective use of finite element codes for practical engineering, analysis. The main problem in the error estimation is the cost of computations and implementing such computations into an existing code structure and hence a fully adaptive finite element structure must be obtained. Various error measures are explained with the help of an example.
Lu-q=S
TDSu-q=0
(16.1)in a domain Ω, with prescribed displacement
u=u
on the boundaryΓ
u (16.2)and prescribed derivatives
Su=t on the boundary
Γ
t (16.3a) with
Γ=Γ
u∪Γ
t (16.3b)where L is linear second-order differential operator, S is first order differential operator, q is a constant vector, D is some matrix and u is primary variable.
In a finite element approximation, we obtain the approximating equations by a standard Galerkin process (or equivalently by minimizing the potential energy) to obtain
Ku-f
=0
(16.4)where
=
∫
( )
Ω
is the elemental stiffness matrix andΩ
SN
D(SN)
d
K
T∫
Ω Ω+∫
Γ = N q N t Γ f T d T d tis the load vector. Here, N is the shape function (interpolation
function) matrix. The approximate solution
u
for an element is related to nodal solution unein
the following manner:
ˆ
uˆ
=N une (16.5)The derivatives are calculated as
σˆ
=(SN)u
ne (16.6) The approximate solutionsuˆ
,σˆ
differs from the exact valuesu
,σ
and the difference is the error. Thus, primary variable and derivative errors are
e=u-u
(16.7a)e
σ= −σ σˆ
(16.7b)The specification of local error in the above equations is generally not convenient and occasionally misleading. For this reason various ‘norms’ representing some integral quantity are
often introduced to measure the error. The most common measures are the ‘energy norm’ and ‘L2 norm’. The energy norm for general problems is
∫
Ω Ω = 2 1 ) ( e Led e T (16.8) wheree=u−uˆ
.A more direct measure is the L2 norm, which can be associated with the errors in any
quantity. Thus for the displacement u, the L2 norm of the error e is
(
)
21
2
=
∫
ΩdΩ
L
e
e
e
T (16.9)and for derivatives,
2 1 ) ) ( ( 2 =
∫
Ω dΩ L σ T σ) e( e eσ (16.10)The ‘root mean square’ (RMS) error for whole domain Ωis given by
2 1 2 2 ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ = Ω e Δu L (16.11)
Similarly for derivatives
2 1 2 2 L
⎛
⎞
⎜
=
⎜
⎟
⎝
⎠
Δσ
Ω
se
⎟ (16.12)
The error for the whole domain is given by summing the element contributions. Thus,
∑
= = m i 1 i 2 2 e e(16.13)
where i represent an element contribution and m is the total number of elements. For an optimal mesh, it is considered that the contributions to the square of the norm is equal for all elements. The relative percentage error
η
can be given by
=
×100%
u
e
16.3 TYPES OF ERROR ESTIMATES
The error estimators for finite element analysis that exist today can be divided into two main categories: a priori error estimates and a posteriori estimates. Finding out the error before the solution is called a priori and after the solution is called a posteriori.
16.3.1 a priori error estimates
In a priori error estimation, the effect of the proposed improved solution is predicted with out actually finding the solution. A priori error estimates provide only a qualitative description on the rate of convergence of the finite element solution. Hence, it is difficult to directly make use of this type of error estimates in a mesh refinement process of adaptive mesh generation, which usually requires a quantitative description of the error distribution as input information. However, this type of error estimates provides an excellent tool for predicting the convergence rate during the adaptive refinement process and also for the estimation of exact energy norms.
16.3.1.1 h – convergence
For h version refinement, the polynomial degree of the interpolation function, p, is kept constant. If the mesh is refined uniformly and the size of the element, h ,approaches zero, the error estimate is given in the form
e
u≤Ch
min(p,λ) (16.15)For 2-D problems, h is approximately proportional to the inverse of the square root of the total degree of freedom of the mesh. Hence,
min( , ) 2 p u
e
CN
λ −≤
(16.16)where N is the total degree of freedom of the mesh,
λ
is the strength of singularities and C is aconstant dependent on the problem but independent of p and
λ
. The mesh is called optimal if thesequence of mesh is designed in such a way that the error is equally distributed in each element and the influence of the singularities is eliminated and
2 p p u
e
Ch
CN
−≤
≈
(16.17)
16.3.1.2 p-CONVERGENCE
For p-version refinement the mesh size is fixed and p is increased uniformly. The error estimate is given by
e
u≤ CN
−β (16.18)where
β
is a positive constant dependent on smoothness of the exact solution and quality of themesh. Hence the rate of convergence will depend very much on the design of the mesh. If a properly designed mesh is used, an exponential rate of convergence can be obtained. If the mesh is not well designed, the performance will be affected especially when singularities are present. In
the case of p-refinement on a uniform mesh in the presence of singularities we have
β
=λ
andthe rate is double than that of the uniform h-refinement. The convergence rate of the p-refinement is always better than that of the uniform h-refinement but care should be taken for the element size near singularities as oscillation of derivatives of solution may occur.
16. 3.1.3 hp-convergence
This simply means that the mesh size, h, is refined simultaneously with the increase of value of p. The error estimate is
( αNϑ)
u Ce
e ≤ − (16.19)
where
α
andϑ
are positive constants dependent on the smoothness of the solution. In practice,there are substantial difficulties in the implementation of the hp-version algorithm as the optimal
mesh for h-refinement depends on p
.
16.3.2 Posteriori error estimates
In posteriori error estimation, the error norms are based on already determined solution and hence the effect of increasing polynomial order or decreasing element size is known and not estimated as is done in a priori error estimation. In addition, because the error is found on a local basis and then summed globally, a posteriori error analyses allow for element level refinement, which is advantageous. They are relatively inexpensive and simple to calculate. Due to the above reasons, a posteriori error estimator has become more popular and was shown to be effective and convergent in many classes of application.
16.3.2.1 ZZ error estimate
The main advantages of using the Zienkiewicz-Zhu (ZZ) error estimator over the other types of error estimators is the simplicity of its implementation and its cost effectiveness. This is
due to the fact that in practical finite element computations some smoothing procedure, which may or may not be superconvergent, will always be employed at the post-processing stage of the computing process to recover the derivatives of the finite element solutions in order to achieve more acceptable approximations. Using such recovered derivatives, the ZZ error estimator can be calculated at a fraction of the total cost of the computation. However, the quality and the reliability of the error estimator is obviously dependent on the accuracy of the recovered solutions and therefore on the smoothing procedures.
To obtain acceptable results for stress, resort is generally made to a nodal averaging or projection
process in which it is assumed that the recovered derivative is interpolated by the same
function as the displacement, i.e.
*
σ
σ* = Nσ* (16.20) where σ*is improved nodal derivative and
(
ˆ)d
Ω
−
Ω =0
∫
N
Tσ
*σ
(16.21) where is elemental FEM derivative.ˆσ
It is intuitively obvious that
σ
*is in fact a better approximation thanσˆ
and we shall use it to estimate the errore
σi.e.* ˆ σ ≈ −
e
σ
σ
(16.22) to evaluate various error norms.16.3.2.2 Residual method
This method is very much useful in p-refinement schemes. In this method, we temporarily introduce a single degree of freedom associated with a shape function of order p+1 into the previous refinement degree of freedom system, where p is the highest order of the shape function present on the element edge and/or face considered. Let us consider that the displacement field of an element is given by,