Errors in FE Modelling (Section 5.10) Modelling error : arises because physical reality is replaced by a mathematical model. Example: A beam that can resist both axial and transverse loads being modelled as one with bending resistance only ignoring the axial resistance. Discretization error : arises because a mathematical model is implemented in a piecewise manner. Example: Perimeter, p, of a circle being computed with a finite number of straight segments. r θ b θ=2π/n ; b=2r sin(θ/2) ; p=nb=2nr sin(π/n) (p/r) exact = 2π. (p/r) n=6 =6 (4.5 %) and (p/r) n=12 =6.212 (1.1 %) (Error shown in parentheses. No modelling error here) 1
Round-off error and ill-conditioning Numerical (round-off) error : arises because a computer uses a finite number of significant digits to represent a number. Such error may cause problem in ill-conditioned systems. Ill-conditioned system : small changes in structural parameters or loading (i.e., in matrix entries) produce large changes in the solution and the solution may be inaccurate. Example : Given a two-dof sytem. Equlibrium equation: 2
Four-Digit Arithmetic Example Say our computer uses 4 digits to represent a number. It would round off numbers with more digits. Let s take k 1 =1.000 and k 2 =4.444e-3 with proper units (four significant digits in each). That is, k 1 >>k 2. Substituting in the two equilibrium equations and writing them out, 1.000(u 1 -u 2 )=P ; -1.000u 1 +(1.004)u 2 =0 Adding the two equations, (1.004-1.000)u 2 =P or (4e-3)u 2 =P Hence, only one significant digit is left! 3
Three-Digit Arithmetic Example With 3-digit arithmetic only, then k 1 =1.00 and k 2 =4.44e-3. The two equations become 1.00(u 1 -u 2 )=P ; -1.00u 1 +(1.00)u 2 =0 which cannot be solved because system is singular. If k 2 were to be changed to 5.00e-3, then 1.00(u 1 -u 2 )=P ; -1.00u 1 +(1.01)u 2 =0 which gives (1e-2)u 2 =P or u 2 =100P. But u 2,exact =200P!!! We see that the round-off error is 50 %. Hence a small change (13%) in a structural parameter (k 2 ) has a very small effect on the stiffness matrix (1%) change, and significant effect on the solution and the solution is inaccurate. 4
Condition With Stiffnesses Interchanged Let s interchange the stiffness values in 4-digit arithmetic so that k 2 =1.000 and k 1 =4.444e-3. That is, k 2 >>k 1. Then, (4.444e-3)(u 1 -u 2 )=P ; -(4.444e-3)u 1 +(1.004)u 2 =0 Adding the two we get 1.000u 2 =P Small changes have only small effects on this solution and the solution is accurate. (Happens to be exact here.) Hence, the equations are well-conditioned. 5
Physical Reason When k 1 >>k 2, u 1 u 2 and spring 1 moves almost like a rigid body. Recall: Presence of rigid-body motion means the global stiffness matrix is singular. When k 2 >>k 1, no rigid-body-like movements and the stiffness matrix is far from being singular and is well-conditioned. Also: When k 1 >>k 2, plotting u 2 vs u 1 for each equation (each row of the equilirium eqn), we see that the curves are almost parallel!! (See Fig. 5.10-1.) 6
Numerical and Physical Ill Conditioning The previous example illustrates ill conditioning of stiffness matrix. A measure of the ill conditioning is the condition number, which is the ratio of the largest and smallest eigenvalues of K. The condition number provides an upper bound on the ratio of relative change in the solution due to relative change in the data. For the two-spring example, the condition number is about 500, so that 0.1% change in a matrix element can change the solution by up to 50%. Physically, the problem is well conditioned, in that small changes in data will cause only small changes in the exact solution. 7
Common causes of ill conditioning The most common source of ill conditioning is great disparity in stiffness between different parts of structure, or even different modes of motion An often self-inflicted ill conditioning is when complex support conditions are replaced by a very stiff support structure. See example in Fig. 5.10-2b Very thin-wall structures may produce this problem because they are very weak in bending compared to stretching Elements with very high aspect ratios, or material properties that are much softer in some directions than others produce the same effect. Large disparities in element sizes also increases ill conditioning even though it is often required in order to focus small elements in regions of high gradients. 8