NUMERICAL SOLUTION
3.13 ERROR TYPES IN THE FINITE ELEMENT SOLUTION
We recall once more that the finite element solution is approximate. This automatically implies that some kind of error in the numerical solution is unavoidable. Next, we will study the more usual sources of error.
3.13.1 Discretization error
This error is intrinsic to the polynomial form of the finite element ximation. We showed in Section 2.7 that the error involved in the appro-ximation is of the order of the first term in the Taylor expansion of the solution not included in the complete shape function polynomial. Strang and Fix [SF] proposed the following general expression to estimate the error for 1D problems
e(error) = uaprox− uexact ≤ Chp+1 Max
¯¯
¯¯∂p+1uexact
∂xp+1
¯¯
¯¯ (3.64) where Max denotes the maximum value of the derivative over the element, C is a constant parameter depending on the element type, h is the ma-ximum characteristic element dimension (i.e. the length in rod elements) and p the degree of the highest complete polynomial contained in the shape functions.
Eq.(3.64) shows that convergence is guaranteed if C and the n + 1th derivative of the solution are bounded. In this case the error will tend to zero as the element size diminishes.
Fig. 3.10 Two rod elements of different sizes
The application of this concept to the 1D linear rod element gives for uniformly spaced meshes
e ∝ h2 ∂2u
∂x2 (3.65)
which implies that the error is proportional to the strain (or stress) gra-dient. Therefore, smaller elements should be used in zones where this gradient is expected to be higher. The reduction of the error by dimini-shing the element size is known in the mesh refinement literature as the h method.
The error can also be reduced by increasing the approximation order of the elements, while keeping their sizes constant. This results in a larger value of the exponent p in Eq.(3.64). This approach is known as the p method.
Eq.(3.64) assumed a mesh of equal element sizes. The effect of using elements of different sizes has been studied for analysis of axially loaded rods using linear elements of two different sizes (Figure 3.10). The error in the satisfaction of the differential equilibrium equation at the jth node is [SF] where h and a are the lengths of two adjacent elements (Figure 3.10).
Eq.(3.66) shows that the error is of the order h2for a uniform mesh (a ' 1), whereas a higher error of order h is obtained when the element sizes are very different (a 6= 1). This indicates that drastic changes in the sizes of contiguous elements in a mesh should be avoided.
The same concepts apply for 2D and 3D problems. The estimation of the discretization error in two dimensions involves the Taylor expansion
u(x + h, y + k) = u(x, y) +
where u is the exact solution and h and k are a measure of the element sizes in the x and y directions respectively.
It can be shown that the discretization error for 2D linear elements, like the 3-noded triangle (Chapter 4), is proportional to the underlined term on the right-hand side of Eq.(3.67) [SF]. The second derivatives in Eq.(3.67) can be related to strain (or stress) gradients. Thus, for a constant strain field the error is very small.
The discretization error can also be expressed in terms of the ratio
kh.This is a measure of the relative dimensions of the element and it is known as the element aspect ratio. For an equilateral element its aspect ratio should be equal to one. However, it will take a large value for a long triangular element. It is recommended to keep the element aspect ratio as close to unity as possible through the mesh.
The estimation of the discretization error will be treated in more detail in Chapter 9, together with the techniques for reducing the error using adaptive mesh refinement.
3.13.2 Error in the geometry approximation
In many cases the interpolation of the geometry is unable to reproduce exactly the real shape of the structure. This can be due to a geometry approximation of a lower order than the exact one, or, what is more usual, to the ignorance of the exact analytical form for the geometry defined by the coordinates of a number of points. In both cases, there will be an error in the geometry approximation. This error can be reduced by refining the mesh, or by using higher order superparametric approximations. A compromise between these two options is to use isoparametric elements.
This unavoidably introduces an error in the geometry approximation in some cases. An exception are structures with linear or planar boundaries where the geometry can always be exactly approximated.
3.13.3 Error in the computation of the element integrals
The exact numerical computation of the element integrals implies using an appropriate quadrature. Otherwise, an error occurs due to the under-estimation of the integral value. In many cases, the exact numerical inte-gration is not possible due to the rational algebraic functions appearing in the element integrals. Also, the approximation of the exact value may require a large number of integration points, which may be very expensive.
In such cases, it is usual to accept a certain error in the computation of the element integrals.
Paradoxically enough, this error can, on occasions, be beneficial. Usua-lly by under-integrating the stiffness matrix terms the element becomes more flexible, and this balances the stiffening introduced by the appro-ximation of the displacement field and the geometry. This explains why sometimes good results can be obtained with coarse meshes. In the fo-llowing chapters we will see that the “reduced integration” quadrature is sometimes used to guarantee the correct solution. The inexact com-putation of the stiffness matrix can however modify its correct rank and introduce spurious mechanisms. Reduced integration is therefore a tech-nique which should be used with extreme care.
3.13.4 Errors in the solution of the global equation system
Three type of errors are typical in the solution of the global system of FEM equations using a direct solution method (i.e. Gaussian elimination, Choleski, Frontal method, etc.): errors due to the ill-conditioning of the equations; truncation errors and round-off errors [Ral].
The equation system Ka = f is ill-conditioned if small changes in the terms of K or f induce large changes in the solution a. The main reason for ill-conditioning is the existence of an element, or a group of elements, of large stiffness connected to elements of much smaller stiffness.
The behaviour of such a structure can be artificially altered and, unless the computer can store a sufficiently large number of digits, the stiffness matrix behaves as singular or quasi-singular. The error associated with ill-conditioning of the equation system therefore depends on the digit storage capacity of the computer, i.e. in the truncation and round-off errors which are the main contributors to the total error in the solution.
The truncation error is quite important. A computer using d digits to represent a number in simple precision can only store the first d digits of each term of K and f. It is then possible that essential information for the correct solution is lost by truncating a number.
The round-off error is due to the adjustment automatically performed by the computer on the last digit of each number during computations.
Experience shows that this error is less important than the truncation error. Nevertheless, unnecessary round-off errors, such as those in some parameters like the coordinates and weights of the numerical quadrature, should be avoided by defining these parameters with the maximum number of digits allowed by the computer.
Fig. 3.11 Spring system with two degrees of freedom
Example 3.7: Study the influence of truncation error in the solution of the spring system shown in Figure 3.11 [CMPW].
-- Solution-- Solution
The system of stiffness equations and its inverse after eliminating the pres-cribed DOF (u3= 0) are the value of the solution. The computation of K−1is only correct if the terms in K are evaluated in a way such that K2is not lost during the solution. Thus, if K1= 80 and K2= 0.0023 the computer must retain at least six digits and K1 must be represented as 80.0000 so that the last digit of K2 is retained in the term K1+ K2. If only four digits are retained the sum K1+ K2 will give 80.00 and K will be singular. This problem is ill conditioned since the solution is sensitive to the changes (truncation) in the sixth digit of the term K1+ K2.
Also, if the system Ka=f is solved using Gauss elimination (Appendix B), the elimination of the displacement u1 changes the last diagonal term to (K1+ K2) − K1. We see that information for a correct solution can again be lost if K1>> K2.
A way to avoid truncation errors and to improve the solution is to use double precision throughout the solution process, i.e. for computing the terms of K and f during the solution of the equation system.
An indicator of how sensitive the system Ka = f is to truncation and round-off errors is the condition number of K. An estimation of the
number of significative figures exactly computed in the solution process is [Bat]
s ≈ t − log10[cond(K)] (3.68) where t is the maximum number of digits which can be stored by the computer and cond (K) is the condition number of K defined as
cond(K) = |λmax|
|λmin| (3.69)
where λmax and λmin are respectively the larger and smaller eigenvalues of K (Appendix A). Although Eq.(3.68) is only approximate, it indicates that the accuracy of the solution decreases as the condition number in-creases.
A low condition number of K is also important in order to speed up the iterative solution of the system Ka = f (Appendix B) [Ral].
It is therefore desirable that the condition number of K should be as low as possible. This can be achieved by an adequate scaling of the terms of K [Ral,RG,RGL].
3.13.5 Errors associated with the constitutive equation
The survey of the error sources in the finite element solution of a struc-ture would be incomplete without referring to the errors arising from a wrong definition of the material properties. In this book only linear elastic materials are considered. The importance of the evaluation of the relevant parameters in the constitutive equation is obvious. For a structure with homogeneous and isotropic material the displacements are proportional to the Young modulus, although the stresses are not affected by this value and they depend only on the Poisson’s ratio. For a structure with or-thotropic or anisotropic materials both the displacements and the stresses depend on the Young modulus and the Poisson’s ratio. We should be aware that an incorrect definition of the material parameters can lead to larger errors than those induced by all the error sources mentioned in the previous sections.