Lifetime-Oriented Structural Design Concepts- P18 docx

468 4 Methodological Implementation 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 ε rel =10 −i ε rel =10 −i ε rel =10 −i ε rel =10 −i εε εε adapt. Simpson-Integration adapt. Combination NC–Gauss adapt. 3-Pt-Gauss-Integration adapt. 4-Pt-Gauss-Integration Fig. 4.66. Strain ε from equation (4.173) for the integral (4.185) 1 2 3 4 20000 40000 60000 80000 100000 1 2 3 4 50000 100000 150000 200000 250000 300000 1 2 3 4 2000 4000 6000 8000 10000 1 2 3 4 5000 10000 15000 20000 25000 30000 1 2 3 4 200 400 600 800 1 2 3 4 500 1000 1500 2000 2500 3000 ε rel =10 −5 ε rel =10 −6 ε rel =10 −3 ε rel =10 −4 ε rel =10 −1 ε rel =10 −2 Int Meth. Int Meth. Int Meth. Int Meth. Int Meth. Int Meth. nn nn nn Fig. 4.67. Number of integration points used in the numerical integration of (4.184) 4.2 Numerical Methods 469 Fig. 4.68. Configuration of a tension test which can be rewritten as f 6 (x, y)= y 32 r 4  2 x 2 (1 + x) 2 − rx(1 + x) 2 y −x (2 + x (7 + 6 x (2 + x))) y 2 − (1 + x) 2 y 4 − y 6  . (4.184) By integrating the function f 6 (x, y) analytically in the integration range x ∈ [−1.02; 1] and y ∈ [−1.01; 1] the integration value  1 x=−1.02  1 y=−1.01 f 6 (x, y) dx = −0.027713772 (4.185) is obtained. The numerical results are shown in Figures 4.66 and 4.67. 4.2.9.2.1.4 p-Version of the XFEM By using the XFEM with bi-linear shape functions to compute the displace- ments of the tension test shown in Figure 4.68 the resulting displacements u x around the crack tip are shown in Figure 4.69 for the deformed system. The black line indicates the asymptotic displacements of the crack surfaces in the vicinity of the crack tip. It can be seen from the figure that the approximation of the displacement field results in big differences to the analytical solution. Furthermore a constriction is noticeable which occurs at the position where the crack cuts the edge of the crack tip element. The results can be improved by using a refinement of the element mesh (h-version or h-extension)aswellashigherorderpolynomialsfortheshape functions (p-version or p-extension). Using a h-extension will improve the results but the constriction is still visible but closer to the crack tip while using a p-version for the shape functions results in the displacements 470 4 Methodological Implementation Fig. 4.69. Displacements u x for the deformed system using bilinear shape functions Fig. 4.70. Displacements u x for the deformed system, left: using bi-quadratic shape functions, right: using quadratic hierarchical shape functions shown in figure 4.70. In both cases the accuracy of the results is improved, but using the hierarchical shape functions the displacement field results in smaller differences and furthermore the constriction is not visible here. To find out more about this phenomenon we examine a single element and compute the coefficients in the XFEM displacement field using the minimizing process of Equation 4.186 n coll  i=1 (u analyt (x i ) −u XFEM (x i )) 2 → min (4.186) 4.2 Numerical Methods 471 The displacements in this equation are the functions of the analytical crack tip field and 4.151. The computed coefficients are used to calculate the integral of the error as defined in Equation 4.187. ε =  Ω |u analyt (x i ) −u XFEM (x i ) |dΩ (4.187) Figures 4.71-4.75 show the results of equation 4.187 for different blending el- ements. Because the condition of the Partition of Unity is satisfied inside the crack tip element the integrated error is zero and not shown here. The errors 1 2 3 4 5 6 7 2 5 lO 7 2 5 lO 6 2 5 lO 5 2 5 lO 4 2 5 lO 3 2 5 lO 2 2 Δu x hierarchic Δu y hierarchic Δu x standard Δu y standard polynomial order  |Δu|dΩ Fig. 4.71. Differences of displacements inside the 1st blending element 1 2 3 4 5 6 7 5 lO 7 2 5 lO 6 2 5 lO 5 2 5 lO 4 2 5 lO 3 2 5 lO 2 2 Δu x hierarchic Δu y hierarchic Δu x standard Δu y standard polynomial order  |Δu|dΩ Fig. 4.72. Differences of displacements inside the 2nd blending element 472 4 Methodological Implementation 1 2 3 4 5 6 7 2 5 lO 8 2 5 lO 7 2 5 lO 6 2 5 lO 5 2 5 lO 4 2 5 lO 3 2 5 lO 2 2 5 0.1 Δu x hierarchic Δu y hierarchic Δu x standard Δu y standard polynomial order  |Δu|dΩ Fig. 4.73. Differences of displacements inside the 3rd blending element 1 2 3 4 5 6 7 2 5 lO 6 2 5 lO 5 2 5 lO 4 2 5 lO 3 2 5 lO 2 2 Δu x hierarchic Δu y hierarchic Δu x standard Δu y standard polynomial order  |Δu|dΩ Fig. 4.74. Differences of displacements inside the 4th blending element for the elements that are not enriched are equal for using Lagrange polyno- mials (the standard shape functions) or Legendre polynomials (hierarchical shape functions). Because of the fact that the shape functions defined for the blending elements span different function spaces, differences for the error of Equation 4.187 exist. The results of the blending elements which are pictured here indicate the enormous influence of the selection of the basis to the approximation of the near tip field. The results show that the blending elements cause a big error of the finite element analysis, because the conditions of the Partition of Unity are not satisfied in the blending elements. The fact that only some of the nodes or 4.2 Numerical Methods 473 1 2 3 4 5 6 7 2 5 lO 6 2 5 lO 5 2 5 lO 4 2 5 lO 3 2 5 lO 2 Δu x hierarchic Δu y hierarchic Δu x standard Δu y standard polynomial order  |Δu|dΩ Fig. 4.75. Differences of displacements inside the 5th blending element modes inside these elements are enriched by the crack tip functions, produces this difficulty. Using the hierarchical shape functions result in smaller errors because the influenced region of the crack tip functions become bigger. For more details on this phenomenon we refer to [618, 334]. 4.2.9.2.1.5 3D XFEM Authored by Christian Becker and G¨unther Meschke Because of the variety and complexity of mechanical engineering crack- propagation problems the powerful two-dimensional X-FEM simulation tools have to be enhanced for three-dimensional problems. In recent years the extended finite element method was succesfully applied to fully three- dimensional problems [781, 547, 780, 301]. The description of the crack topol- ogy and crack propagation is more frequently described implicitly by the level set method [606, 605]. Considering the three-dimensional extended finite element method, there is mostly a conflict between the accuracy, applicability, realistic results and the complexity of the numerical implementation, in particular regarding crack propagation criteria. As an example, in a two-dimensional analysis of a single crack, the crack front is represented by a single point P . Consequently, the crack propagation is simply reduced to the determination of the crack prop- agation angle Θ c and the crack propagation length l c .Evenwithacrack propagation criterion like the minimization of the total energy, this problem is solved quite easily by introducing two additional system degrees of freedom Θ c and l c . On the contrary, in a full three-dimensional analysis of crack propagation of a single crack, the crack front is represented by a number of line segments bounding the crack surface. It can be seen, that this problem is of much more complexity than the two-dimensional problem. 474 4 Methodological Implementation In most of the three-dimensional X-FEM implementations tetrahedral ele- ments with linear approximations of the displacement field are used in combi- nation with elementwise plane crack propagation (see e.g. [52]). This represents the classical h-finite element approachto capture the crack propagationprocess. A more complex simulation concept applies the well-known level-set method (see e.g. [546, 324]). Within this method arbitrary crack growth is accounted for by solving Hamilton-Jacobi-like equations. A crucial point in this con- cept is that the velocity of the crack surface evolution has to be known to capture the propagation process. Alternatively, the proposed combination of the p-FEM and the X-FEM that was stressed in the last section (Section 4.2.9.2.1.4) and holding for high accuracy, is applied to the numerical simulation of three-dimensional crack propagation. In this concept, because of the complexity of the fully three-dimensional problem, some assumptions are introduced. The underlying finite element formulation is a continuum brick element holding for arbitrary higher-order shape functions (see Section 4.2.4.3). The proposed higher order X-FEM simulation strategy holds for: • element-wise crack propagation • plane crack surface at the element level With the proposed assumptions a row of questions arise concerning numerical integration and the determination of crack propagation. Because of the jump in the enhanced strain field, standard Gauss- Integration cannot be applied. Hence, both parts of a cracked element have to be integrated separately. In two dimensional analyses the well-known De- launey-triangulation is used. A division into sub-tetrahedra in the sense of the Delauney-triangulation is way too complicated, therefore a fixed sub- division into six subtetrahedra is used (see Figure 4.76). Each of these six tetrahedra may be cut by the crack plane into the sub-domains of a pentahe- dron, tetrahedron or a pyramide by either a triangular or quadrilateral crack plane (see Figure 4.77). The element quantities (•) like stiffness matrices or in- ternal load vectors are generated by summing over all obtained sub-domains that are numerically integrated with a Gauss-integration according to the geometrical shape of the sub-domain V i η : (•)=  Ω f(X) dΩ =  V ξ f(X(ξ)) || ∂X ∂ξ ||dV ξ = n  i=1  V i η f(X(ξ(η))) || ∂X ∂ξ ∂ξ ∂η ||dV i η . (4.188) Besides numerical integration, the proposed assumptions lead to a problem concerning crack propagation, that is adressed in the following paragraph. In general, the crack path continuity condition is fullfilled in the context of the 4.2 Numerical Methods 475 Fig. 4.76. Numerical integration in the context of the X-FEM: Subdivision of the continuum element into six sub-tetrahedrons Ω − Ω + ∂ S Ω ✟ ✟ ✟✯ n Ω − Ω + ∂ S Ω   ✒ n Fig. 4.77. Separation of a sub-tetrahedron by a plane crack segment: left: separa- tion into two pentahedra by a quadrilateral, right: separation into pentahedron and tetrahedron by a triangle X-FEM. In conjunction with elementwise, plane crack propagation this leads to strong restrictions concerning the kinematics of the evolving crack surface. These restrictions are addressed in Figure 4.78. For a start, the crack front is represented by line segments at those element faces where the crack surface ended so far. Sound elements that are neighbours to that faces are crack candidates and are investigated in the subsequent load 476 4 Methodological Implementation Ä P 1 P 2 P 3 P 1 P 2 X 1 X 3 P 1 P 2 n n mod Fig. 4.78. C 0 -crack plane evolution: Left: fixed position of the potential crack segment by two neighbouring crack tip line segments, middle: restricted position of the crack segment by one neighbouring crack tip, right: modification of the computed normal vector considering one neighbouring crack tip segment steps if the crack propagates through them. In Figure 4.78 the element in the middle is neighboured by only one crack segment whereas the candidate at the left is bounded by two segments of the crack front. Because of the already existing crack line of the middle element, the position of the new crack segment is not fully arbitrary but can only be positioned by a rotation around the existing crack line segment. Therefore the normal vector resulting from the crack propagation criterion has to be modified, like it is illustrated at the right hand side of Figure 4.78. Regarding the left element, the position of a crack segment is already predefined by the two linear independent line segments. Therefore, it is only possible to decide when the new crack plane is inserted. The preceding thoughts occur only in the context of a C 0 -continuous description of the crack surface, using an algorithm according to e.g. [52]. To gain more flexibility concerning the kinematics of the evolving crack sur- face, the C 0 -continuity of the crack surface is neglected and a C 0 -discontinuous algorithm according to [301] can be used. In this algorithm the new crack seg- ment introduced within a crack candidate is defined by a point P and the normal vector n obtained through the crack propagation criterion. The point P is the geometrical mean of all existing mid-points of existing crack segments at the el- ement boundaries (see Figure 4.79). With this method the crack propagation gains a lot of flexibility by keeping the numerical implementation very simple. In cases where the gaps between neighboured crack segments are getting to wide, a smoothing algorithm for the crack surface is provided by [302]. 4.2.9.2.1.6 XFEM for Cohesive Cracks Authored by Christian Becker and G¨unther Meschke First of all, the X-FEM was applicated to problems of Linear Elastic Fracture Mechanics (LEFM). Therein, the general balance of momentum 4.2 Numerical Methods 477 P 1 P 2 P Fig. 4.79. Definition of the crack plane by point P and normal vector n. P is the geometric mean of all midpoints of the crack front lines at the discontinuity, requiring the jump of the traction to vanish at the discontinuity [[ t d ]] = 0 , (4.189) was accounted for by stating vanishing traction t d =0. (4.190) LEFM is primarily used in the context of numerical simulations of brittle materials like ceramics. Numerical simulations of quasi-brittle material behaviour of concrete require the consideration of a cohesive zone by a cohesive zone modell [240, 85, 369]. Within the cohesive zone the traction vector is not vanishing, but is dependend on the energetically conjugated variable of the displacement jump [[u]] : t d = t d ([[u]] ) . (4.191) Because of the non-vanishing traction vector, there is an additional term of the internal virtual work within the weak form of balance of momentum, here, for brevity without volume or external loads:  Ω δ ¯ ε : σ dΩ +  Γ d δ[[ u]] · t d dΓ d    δW int =0. (4.192) Exemplary, an anisotropic softening law according to [828] is investigated. The hyperbolic softening is realized by application of a scalar damage criterion [...]... identity ∗ ϕ(X) = N∗ (X) (4.211) holds (see [598]) Originally, the ramp function (4.211) was proposed for the design of a numerical length scale, cf [596] The extension to higher order elements is straightforward Applying the interpolation conditions and considering the most general case, ϕ is designed according to nΩ + ¯ ϕ= Ni (4.212) i=1 nΩ + ¯ Here and henceforth, the notation denotes the summation... post-peak regime The resulting structural response is given in Figure 4.83 Finally, a numerical analysis based on a multiple fixed crack model is performed, i.e., two orthogonal cracks per element are allowed which do not rotate ˙ (N = 0) Analogously to the rotating SDA, no convergence problems occur if intersecting cracks are allowed However, as illustrated in Figure 4.83, the structural response computed... 4 Methodological Implementation Numerical Implementation Since the re-formulated SDA as described in the previous subsection is formally identical to classical plasticity theory, algorithms originally designed for standard plasticity models can be applied with only minor modifications necessary In this section, an adapted return-mapping algorithms is briefly presented, cf [743, 741] Further details may... models Therefore, the by now well-known return mapping algorithm [743, 741] can be applied to solve the resulting nonlinear set of differential equations Only minor modifications of material subroutines designed for standard plasticity models are required Clearly, this represents an important advantage of the finite element formulations proposed in [139, 557, 558] compared to the original one presented . the design of a numerical length scale, cf. [596]. The extension to higher order elements is straightforward. Applying the interpolation conditions and con- sidering the most general case, ϕ is designed. resulting nonlinear set of differential equations. Only minor modifications of material subroutines designed for standard plasticity models are required. Clearly, this represents an important advantage. previous subsection is formally identical to classical plasticity theory, algorithms originally designed for standard plasticity models can be applied with only minor modifications necessary. In