Lifetime-Oriented Structural Design Concepts- P17 doc

438 4 Methodological Implementation 0 10 20 30 40 50 60 70 80 -700 -600 -500 -400 -300 -200 -100 0 load level [-] w [mm] Beam 1 (nGP=16, crit2) tolerr=0.1 tolerr=0.01 tolerr=0.005 tolerr=0.001 Q1SPs 10 20 30 40 50 60 70 80 90 100 -700 -600 -500 -400 -300 -200 -100 0 number of elements [-] w [mm] Beam 1 (nGP=16, crit2, Q1SPs/o) tolerr=0.1 tolerr=0.01 tolerr=0.005 tolerr=0.001 Fig. 4.34. Beam 1: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs/o, nGP0 = 16) Fig. 4.35. Beam 2: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs/o, nGP = 16) increase the number of elements up to 67 (Figure 4.35b). The curve of Q1SPs computed with a very small tolerance (see the curve s : tolerr = 0.00001 in Fig- ure 4.35a) hardly differs from the result achieved with a rather large tolerance (Figure 4.35a, s : tolerr = 0.05). Different states of mesh refinement for a computation with Q1SPs/o are plotted in Figure 4.36. As expected refinement starts in the loading area and in the parts of the structure where the plastic strain begins to accumulate (X ≈ 4500 mm). In conclusion it should be emphasized that also the error criterion crit2 provides a physically reasonable mesh refinement. Additionally, in contrast to crit1, it measures the quality of the solution accurately enough, i.e. in such a way that computations based on a too stiff element formulation (e.g. Q1SPs/o) require a much higher mesh density for the same quality of solution. Therefore 4.2 Numerical Methods 439 E2: 0.000648603 0.0705965 0.140544 E2: 0.000648603 0.0705965 0.140544 E2: 0.000648603 0.0705965 0.140544 E2: 0.000648603 0.0705965 0.140544 Fig. 4.36. Beam 2: Different states of mesh refinement (Q1SPs/o, 16 El.), contours: accumulated plastic strain the mesh refinement in Figure 4.36 is here shown for Q1SPs/o. Q1SPs shows the same mesh refinement when the error tolerance is set equal to an extremely small value. 4.2.8.1.10.3 Biaxial Bending (Thick Plate of Uniform Thickness) Geometry and boundary conditions are depicted in Figure 4.37. At the boundary X = 12000mm the displacement in X-direction is constrained. Analogously at the boundary Y = 12000mm we apply constraints in Y - direction. The material parameters and the reference load are the same as in the previous example. The maximum load level is ν max = 60 (applied in 24 equidistant steps). For this example also Q1SPs requires a refinement of the mesh in thick- ness direction to yield a converged result (see Figure 4.38a). The solution for tolerr =0.0001 (refinement up to 140 elements, see Figure 4.38b) is very close to the one computed with a mesh of 7x7x4 elements. The curves for the larger tolerances (tolerr ≤ 0.0002) bring up the problem that at the beginning of the refinement (one to two new elements) the solution stiffens, i.e. the displace- ment w becomes smaller, see the detail A in Figure 4.39. At a larger load the solution softens again (detail B in Figure 4.39). However, the solution for tolerr =0.001 must be judged as worse as the result for tolerr =0.01 where 440 4 Methodological Implementation X Y Z 1200 mm 12000 mm w 12000 mm F F Fig. 4.37. Plate 1: Geometry and boundary conditions 0 10 20 30 40 50 60 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 load level [-] w [mm] Plate 1 (nGP=8, crit2, Q1SPs) tolerr=0.01 tolerr=0.001 tolerr=0.0005 tolerr=0.0002 tolerr=0.0001 tolerr=0.00001 nGP=8 maximum 40 60 80 100 120 140 160 180 200 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 number of elements [-] w [mm] Plate 1 (nGP=8, crit2, Q1SPs) Fig. 4.38. Plate 1: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs, nGP =8) no mesh refinement takes place. We assume that the problem is due to the transfer of the history variables which can never be exact. The element formu- lation Q1SPs reacts much more sensitively to this transaction than Q1SPs/o. The deficiency is easily overcome by working with a slightly smaller tolerance, where right at the beginning several new elements are inserted. In summary it can be, however, stated that the differences between the curves for rather large and very small tolerances are again small. The picture is as expected very different if one uses Q1SPs/o (see Figure 4.39). Whereas Q1SPs does not need additional elements to undercut the error tolerance tolerr =0.01, Q1SPs/o requires for a solution of the same quality a refinement up to 148 elements. Applying the tolerance tolerr =0.001 leads already to the maximum refinement (196 elements), i.e. with nGP =8 the solution cannot be further improved. It has still not completely converged at this point. 4.2 Numerical Methods 441 0 10 20 30 40 50 60 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 load level [-] w [mm] Plate 1 (nGP=8, crit2, Q1SPs/o) tolerr=1.0 tolerr=0.1 tolerr=0.01 tolerr=0.001 0 10 20 30 40 50 60 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 load level [-] w [ mm ] Plate 1 (nGP=8, crit2, Q1SPs) tolerr=0.01 tolerr=0.001 nGP=8 maximum A B Fig. 4.39. Plate 1: Load-displacement curve (left) and details of it (right) for dif- ferent tolerances and crit2 (Q1SPs, nGP =8) Fig. 4.40. Plate 1: Different states of mesh refinement (Q1SPs/o, 16 El.), contours: accumulated plastic strain The adaptive increase of the number of elements is visualized in Figure 4.40 (computation with Q1SPs/o). The mesh refines strongly in the loading domain where a severe element distortion is detected. At this point a simultaneous refinement in the plate plane is necessary. It should be made available in the future. 442 4 Methodological Implementation Fig. 4.41. Plate 1: Load-displacement curve and number of elements for different load steps and crit2 (Q1SPs/o, nGP =8,tolerr =0.01) 0 10 20 30 40 50 60 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 load level [-] w [mm] Plate 1 (nGP=8, crit2, Q1SPs, tolerr=0.0001) dF=2.5 dF=5.0 dF=10.0 dF=20.0 dF=30.0 dF=60.0 40 50 60 70 80 90 100 110 120 130 140 150 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 number of elements [-] w [mm] Plate 1 (nGP=8, crit2, Q1SPs, tolerr=0.0001) dF=2.5 dF=5.0 dF=10.0 dF=20.0 dF=30.0 dF=60.0 Fig. 4.42. Plate 1: Load-displacement curve and number of elements for different load steps and crit2 (Q1SPs, nGP =8,tolerr =0.0001) A point of further interest is the robustness of the algorithm. For this purpose we compare first for Q1SPs/o and tolerr =0.01 the load-displacement curves computed with different load steps (see Figure 4.41). We obtain the quite impressive result that the total load (ν =60)canbeappliedinone step which shows the robustness of the proposed algorithm. The resulting displacement is only marginally smaller than the correct value (for tolerr = 0.01). The new elements appear in almost the same order as for much smaller load steps. The same robust behaviour is observed for the computation with Q1SPs and tolerr =0.0001. However, the number of new elements is in in this case influenced by the load step. Fewer new elements are generated if the load step is chosen to be very large. Moreover the solution shows the peculiar stiffening again, i.e. the insertion of a few new elements does not lead as expected to a softer (in this case better) solution. 4.2 Numerical Methods 443 4.2.8.2 Error-Controlled Temporal Adaptivity Authored by Detlef Kuhl and Sandra Krimpmann Strategies for the error controlled adaptive time integration of durability problems consist of to main parts: The computation of error measures and the control algorithm of the time step size. Local error estimates and different local error indicators can be used as controlling parameter in adaptive time stepping schemes. 4.2.8.2.1 Local a Posteriori h-andp-Method Error Estimates In order to obtain an estimate of the local time integration error, error mea- sures based on the comparison of the primary integration results u n+1 with simultaneously performed time integrations of higher accuracy are used. As illustrated in Figure 4.43 for Newmark and Galerkin type integration schemes the integration quality can be improved by reduction of the time step size Δt. Reasoned by the fact that the time integration error e ∼ Δt o is proportional to the time step size Δt to the power of the order of accuracy o, the error e can be significantly reduced if the time step size is divided by m. The resulting improved solution u Δt/m n+1 allows for the estimation of the local time integration error by the h-method. e Δt/m = u n+1 − u Δt/m n+1 (4.136) The order of accuracy of Galerkin type integration schemes is controlled by the polynomial degree p whereby the higher the polynomial degree the higher h-method error measures p-method error measures error estimates error indicators enlarged time step size basic time steps decreasedtimestepsize t Δt u Δt n−1 u Δt n u Δt n+1 u Δt n+2 e Δt/m e Δt/m e Δt/m e mΔt Δt m u Δt/m n u Δt/m n+1 u Δt/m n+2 mΔt u mΔt n+2 polynomial degree p 1 = p −m<p polynomial degree p 2 = p + m>p t Δt u Δt n−1 u Δt n u Δt n+1 u Δt n+2 e p/p 2 e p/p 2 e p/p 2 e p/p 1 e p/p 1 e p/p 1 u p 2 n u p 2 n+1 u p 2 n+2 u p 1 n u p 1 n+1 u p 1 n+2 Fig. 4.43. Illustration of h-method error estimates and indicators associated with Newmark and Galerkin time integration schemes as well as p-method error esti- mates and indicators for p-Galerkin time integration schemes 444 4 Methodological Implementation the order of accuracy. For p-Galerkin type integration schemes a higher order accurate comparison solution can be alternatively generated by Galerkin integration schemes of a higher polynomial degree p + m or a Galerkin integration with a smaller time step size Δt/m, compare Figure 4.43. The resulting improved solution, denoted by u Δt/m n+1 , allows for the estimation of the local time integration error by the h-method. The resulting improved solution u p+m n+1 allows for the estimation of the local time integration error by the p-method. e p/p+m = u p n+1 − u p+m n+1 (4.137) Both present error estimates of the h-andp-method are excellently appro- priate to estimate the real time integration error. This exceeding quality of the error measures enforces, of course, a very high numerical effort. As a con- sequence of this, these error estimates are applied if highly robust adaptive integrations and absolutely reliable calculations of multiphysics problems are necessary. Since unreliable prognoses of the long term behavior of concrete structures are worthless, it is highly recommended to invest the additional computational time for the error estimates of the h-andp-method. If the solution behavior of the durability problem is completely understood by the engineer he may switch to the error indicators of the h-andp-method for further parametric studies. Furthermore, these error estimates are applied to study the numerical properties of the present time integration schemes in the context on non-linear durability problems. 4.2.8.2.2 Local a Posteriori h-andp-Method Error Indicators Error indicators of the h-andp-method are only applied, if the numerical effort for the error estimates discussed in the previous section is significantly to high. These kind of error indicators are characterized by comparison so- lutions of lower quality. The present error indicators are either based on the h-method e mΔt = u n+1 − u mΔt n+1 (4.138) or the p-method, compare Figure 4.43. e p/p−m = u p n+1 − u p−m n+1 (4.139) 4.2.8.2.3 Local Zienkiewicz a Posteriori Error Indicators Motivated by the high numerical effort of h-andp-method error estimators as well as indicators, alternative error measures using the Taylor expansion of the solution u n+1 for the comparison with the Newmark solution were developed. This basic idea was firstly published by [868] and later enriched by several extensions by [494, 675, 833, 834, 566]. So called Zienkiewicz error 4.2 Numerical Methods 445 Table 4.6. Error indicators for Newmark type time integration schemes (e ZX : [868], e LZW : [494], e RS : [675], e R : [676]) for non-linear second order initial value problems r i ( ¨ u, ˙ u, u)=r derivative ˙ ¨ u n derivative ¨ ¨ u n error indicator e in e ZX 1 Δt [ ¨ u n+1 − ¨ u n ] 0 6β−1 6 [ ¨ u n+1 − ¨ u n ]Δt 2 e LZW 2 Δt [ ¨ u n+1 − ¨ u n ]− ˙ ¨ u n 1 Δt [ ˙ ¨ u n+1 − ˙ ¨ u n ] 12β−1 12 [ ¨ u n+1 − ¨ u n ]Δt 2 − 1 12 ˙ ¨ u n Δt 3 e RS 1 2Δt [ ¨ u n+1 − ¨ u n−1 ] 1 Δt 2 [ ¨ u n+1 −2 ¨ u n + ¨ u n−1 ]  8β−1 8 ¨ u n+1 + 1−12β 12 ¨ u n + 1 24 ¨ u n−1  Δt 2 e R 0 0 β[ ¨ u n+1 − ¨ u n ]Δt 2 indicators compare the Taylor expansion of the primary variable at the end ofthetimestep u ZX n+1 = u n + Δt ˙ u n + Δt 2 2 ¨ u n + Δt 3 6 ˙ ¨ u n + Δt 4 24 ¨ ¨ u n + ··· (4.140) and the Newmark approximation. e ZX = u n+1 − u ZX n+1 = β γ [ ˙ u n+1 − ˙ u n ]Δt − 1 2 ¨ u n Δt 2 − 1 6 ˙ ¨ u n Δt 3 − 1 24 ¨ ¨ u n Δt 4 −··· (4.141) Table 4.6 summarizes various existing error indicators developed for second or- der initial value problems on basis of equation (4.141). They are distinguished by the estimation of the higher order time derivatives ˙ ¨ u n and ¨ ¨ u n .Namely, • the Zienkiewicz Xie error indicator, • the Li Zeng error indicator, • the Riccius error indicator, and • the Rickelt error indicator are special cases of the error indicator defined by equation (4.141). Since this er- ror indicators are defined in terms of accelerations different derivations for first order initial value problems compared to second order initial value problems on the basis of equation (4.141) are required. They substitute ¨ u n and ˙ ¨ u n by ade- quate difference approximations in terms of velocities, see Table 4.7. 446 4 Methodological Implementation Table 4.7. Error indicators for Newmark type time integration schemes (e ZX : [868], e LZW : [494], e RS : [675], e R : [676]) for non-linear first order initial value prob- lems r i ( ˙ u, u)=r derivative ¨ u n derivative ˙ ¨ u n error indicator e in e ZX 1 Δt [ ˙ u n+1 − ˙ u n ] 0 2β − γ 2γ [ ˙ u n+1 − ˙ u n ]Δt e LZW 2 Δt [ ˙ u n+1 − ˙ u n ] − ¨ u n 1 Δt [ ¨ u n+1 − ¨ u n ] 3β − γ 3γ [ ˙ u n+1 − ˙ u n ]Δt − 1 6 ¨ u n Δt 2 e RS 1 2Δt [ ˙ u n+1 − ˙ u n−1 ] 1 Δt 2 [ ˙ u n+1 −2 ˙ u n + ˙ u n−1 ] Δt  12β−5γ 12γ ˙ u n+1 + γ −3β 3γ ˙ u n + 1 12 ˙ u n−1  e R 0 0 β γ [ ˙ u n+1 − ˙ u n ]Δt 4.2.8.2.4 Adaptive Time Stepping Procedure As a basis of adaptive time stepping procedures the error vectors (4.136,4.137, 4.141) are transformed to a scalar valued relative error measure by using various alternative reference values u ref : e = e u ref u ref = u n+1 − u n  u ref = u n+1  u ref = u 0  (4.142) The error measure e is compared with the user defined error bounds ν 1 η and ν 2 η. ν 1 η ≤ e ≤ ν 2 η (4.143) If equation (4.143) is fulfilled the time step remains unchanged. Otherwise, the time step will be adapted: Δt new = Δt old o  η e (4.144) o represents the order of accuracy of the basis time stepping scheme. For e>ν 2 η the last time step is repeated with Δt new and for e<ν 1 η the next time step is solved with Δt new . 4.2 Numerical Methods 447 loop over time steps n n +1→ n initial conditions Newmark-α time integration (Figure 4.24) ¨ u n+1 , ˙ u n+1 , u n+1 indication of error e in = u n+1 − u in n+1 scalar valued relative error measure e = e in /u ref error check (y|n) ν 1 η ≤ e ≤ ν 2 η retain time step size Δt calculation of time step size Δt new error check (y|n) ν 1 η>e next time interval [t n+1 ,t n+2 ] −→ [t n ,t n+1 ] retry time interval [t n ,t n+1 ]final update of internal variables κ Fig. 4.44. Algorithmic set-up for the error controlled adaptive time integration by Newmark-α integration schemes combined with error indicators and the adaptive time step control by [868], compare Figure 4.24 loop over time steps n n +1→ n loop over time steps m m +1→ m initial conditions integration u mΔt m+1 , u Δt/m m+1 standard integration by Newmark-α or p-Galerkin methods u n+1 (Figures 4.24 and 4.26) indication/estimation of error e mΔt = u n+1 − u mΔt n+1 , e Δt/m = u n+1 − u Δt/m n+1 scalar valued relative error measure e = e/u ref error based adaptive time step control (Figure 4.44) Fig. 4.45. Algorithmic set-up for the error controlled adaptive time integration by Newmark-α or p-Galerkin methods and h-method error estimates/indicators 4.2.8.2.5 Algorithmic Set-Up A typical algorithmic set-up for the adaptively controlled time integration of non-linear second order semidiscrete initial value problems is shown in Figure 4.44. In this overview the Newmark type integration scheme by [569, 198] is combined with error indicators and the adaptive time stepping procedure by [868] as a representative example. The boxes in Figure 4.44 illustrate links with the element and material levels of multifield durability finite element programs (compare Section 4.2.7). As illustrated by Figures 4.45 and 4.46