to quantum chemistry, in which the solutions present some singularities at the position of the nuclei. To build the reduced basis, the solutions were expressed somehow “centered” around these nuclei [CAN 07]. A similar strategy was also performed in the context of parameterized domains, solving for the steady Stokes problem. The solutions were mapped to a reference domain in order to compute the reduced basis shape functions properly [LØV 06].
From a general point of view, coupling a classical model reduction approach and the morphing technique constitutes in fact anonlinear dimensionality reductiontechnique [LEE 07,FOD 02].
Indeed, it is somehow similar to the so-called Kernel PCA strategy [SCH 97]: the snapshots are obtained in an observation space, in which they cannot be approximated by a low dimensional lin- ear subspace. Then in a second time they are mapped to a feature space in which a classical PCA is conducted more successfully. The nonlinear dimensionality reduction (NLDR) techniques, also called manifold learning methods, are innovative tools widely used in the fields of data analysis, data mining and machine learning. They aim at providing a way to understand and reveal the structure of complex data sets. Indeed, it is clear that sometimes the traditional methods such as PCA do not success in unfolding the intrinsic dimension of a set because they are based on linear models. On the contrary, the NLDR methods are relatively recent since they emerged in the late 1990’s, and treat the data in a nonlinear way, opening new possibilities. To the best of our knowledge, for the moment, those methods have not been applied as such in the field of reduced basis and model reduction. In this respect they might deserve to be studied a little more.
To finish with this comparison of the node release technique versus the morphing approach, some of the obtained SVD modes are presented on Figure2.25for the node release approach and on Figure2.26for the mesh morphing technique. Remember that the displacement fields were computed on different meshes, according to the crack propagation. Here, the shape functions are presented projected on the initial mesh although only the first one was computed on this mesh.
As a matter of fact, for both techniques, the very first modes of the SVD are able to capture the global characteristics of the displacement solutions corresponding to the whole propagation simulation. After that, the subsequent modes are providing more local information about the crack tip fields. In general, the higher the order of a shape function, the more localized on the front its mode. It is apparent that the higher order shape functions obtained on the model which uses the mesh morphing technique contain more artifacts and perturbations than the corresponding shape functions of the model with the node release strategy. However, in the present situation, this does not constitute a real problem since the SVD modes to which they are associated correspond, in the morphing case, to singular values that are less than 10−7. This means that those shape functions would have a very low influence on the solutions that would be computed using their associated reduced basis, and even that, in practice, they would probably not be retained in the reduced basis.
Nonetheless, some hints to explain this phenomenon will be introduced in the next section.
Model reduction and fracture mechanics: discussion around the mesh morphing technique 99
(a) First SVD mode (b) Second SVD mode
(c) Third SVD mode (d) Fourth SVD mode
(e) Fifth SVD mode (f) Sixth SVD mode
Figure 2.25 Six first modes of the singular value decomposition of the displacement solution vectors computed with the node release approach.
(a) First SVD mode (b) Second SVD mode
(c) Third SVD mode (d) Fourth SVD mode
(e) Seventh SVD mode (f) Eighth SVD mode
Figure 2.26 Some of the first modes of the singular value decomposition of the displacement solution vectors computed with the mesh morphing technique. The modes are presented projected on the initial mesh.
Model reduction and fracture mechanics: discussion around the mesh morphing technique 101
Φ10
50
Figure 2.27 Geometry, in millimeters, of the crack in a cylindrical sample.
(a) (b) (c) (d)
Figure 2.28 Norm of the displacement solution for respectively the initial(a)and final(b) configurations, and associated von Mises stresses(c)–(d).
this cylindrical tension sample are provided on Figure2.28(a). The same results are also presented on the cross-section of the sample located on the crack plane Figure2.30.
As a matter of fact, the general shape of the solution fields on initial and final configurations are fairly similar, and anyway not much different that the initial and final fields of the rectangu- lar tension sample are (section6.2). Consequently, one would reasonably infer that the intrin- sic dimension of the solution manifold for that cylindrical beam test-case should not be much higher than the one of the rectangular sample. For all that, the singular value decomposition of the cylindrical sample snapshot matrix is computed, and an orthonormalization of the snapshots is conducted by means of a Gram-Schmidt process as well. The resulting normalized singular values, and the norm of the Gram-Schmidt basis vectors are plotted on Figure2.29. As a mat- ter of fact, the singular values are stagnating around 10−5, and the norm of the Gram-Schmidt shape functions around 10−4, meaning that the information contained in those snapshots cannot be compressed, nor approximated by a space of low dimension. A visual manifestation of this phenomenon can also be provided, plotting the shape functions resulting from the Gram-Schmidt process (see Figure2.31). The vectors are presented projected on the initial mesh. The shape functions do not localize around the crack front, as it was the case for all the previous test-cases presented in this typescript. Rather, they are quite irregular and artifacts appear, enriching the basis in zones far from the crack.
A first hint to explain these deceptive results is coming from the analysis of the non-linear mappingsχi of Equation (2.59). Those are the straightforward mappings provided by the mesh morphing allowing to map the displacement solution vectors, and by extension the reduced basis
1e-006 1e-005 0.0001 0.001 0.01 0.1 1
20 15
10 5
1
Value
Singular value number / Vector number Normalized singular value Gram-Schmidt process residue
Figure 2.29 Normalized singular values of the set of 20 successive displacement solution vectors and residual of the Gram-Schmidt process during their orthogonalization.
shape functions as well, from the configuration on which they were computed to the current configuration of the cracked sample. In fact, this mapping approach is very simple since it only consists in letting the nodal values of the solution vectors be expressed without any modification on all the successive deformed meshes. However, its major drawback is that it does not ensure that the mapped field exhibit thesame characteristicsas the initial ones. To put it another way, it does not preserve important features such as the divergence or the orientation of the vector field with respect to the external boundaries of the studied body. Indeed, the solution fields of a specific geometric configurationΩobelong to its associated solution spaceU(Ωo). An efficient mapping should hence guarantee that the transformed fields to the current configurationΩcare projected onto its associated solution spaceU(Ωc), which is not the case with the method proposed in this typescript. A solution to alleviate this type of difficulty is to choose a more appropriate, yet more complex mapping, such as for instance a mapping based on the Piola transformation, as it has been pointed out in a some papers [LØV 06,DEP 10]. However, it has also been shown that computing precisely a discrete Piola transformation is particularly complex.
A second hint appears through a refined analysis of the solution displacement fields evolutions along the propagation. For that purpose, several vector plots of these displacement fields are pro- vided on Figure2.32. They are presented on the cross-section of the sample located on the crack plane, and the size of the plotted vectors is scaling with the magnitude of the displacement fields.
First, the displacement solution of initial crack configuration is plotted on its corresponding mesh (see Figure2.32(a)). The displacement vectors are mainly pointing toward the center of the sam- ple. This is basically the expected behavior of a cylindrical sample under tension with a small crack or even without any crack. Indeed, because of the Poisson effect, the displacement field is not oriented only along the longitudinal direction, but also toward the center of the specimen.
In the symmetry plane of the sample, only the radial component is visible. After that, the same initial displacement field ismappedto the final crack configuration and shown on the final mesh (Figure2.32(b)). In spite of the naive mapping that has been used, the initial field can be easily recognized. This mapped vector constitute in fact the first of the shape function that would be used in a reduced basis approach to compute the displacement solution of the final configuration as a linear combination of the previous solution vectors. Finally, the displacement solution corre- sponding to the final crack position is presented on the final mesh on Figure2.32(c). Clearly, the characteristics of the shape function of Figure2.32(b)is not matching those of the final solution
Model reduction and fracture mechanics: discussion around the mesh morphing technique 103
(a)
(b) (c)
(d) (e)
Figure 2.30 Intensity of the von Mises stresses for respectively the initial(a)–(b)and final(c) configurations, and norm of the displacement solution(d)–(e). The results are presented on the cross-section of the sample located on the crack plane.
(a) First shape function (b) Second shape function (c) Third shape function
(d) Fourth shape function (e) Fifth shape function (f) Sixth shape function
Figure 2.31 First shape functions obtained by applying a Gram-Schmidt process on the displacement solution vectors of the round bar test-case. The shape functions are presented projected on the initial mesh.
(a) (b)
(c)
Figure 2.32 Displacement vectors presented on the cross-section of the sample located on the crack plane. The size of the vectors is scaling with the magnitude of the displacement. Initial solution presented on the initial crack configuration(a). Initial solutionmappedto the final configuration(b).
Final solution presented on its corresponding final crack configuration(c).
Model reduction and fracture mechanics: discussion around the mesh morphing technique 105
(a) (b)
(c) (d)
Figure 2.33 Displacement vectors presented on the cross-section of the samples located on the crack plane. The size of the vectors is scaling with the magnitude of the displacement. The results are presented for both the rectangular tension sample test-case, and the Tada’s crack case. First, the ini- tial solutions are presented on their corresponding initial crack configurations(a)–(c), then the final solutions are shown on their corresponding final configurations(b)–(d).
field. Indeed, the grown crack had an effect on the whole displacement field distribution, and the vectors are no more pointing toward the center of the sample, but rather toward the crack itself. In fact, during its propagation, the crack had alarge scaleeffect on the solution field. The evolving crack generates a global effect on the structure. This means that, in this problem, there is no
“separability” of the scale of the structure and the scale of the crack. The reduced basis method is hence perturbed by this observation.
In fact, reaching the latter conclusion would have been a trivial task if it would have sufficed to compare a characteristic length of the crack to a characteristic dimension of the structure (the cylinder radius for instance) in order to predict the reduced basis bad behavior. Obviously, the larger the crack compared to the scale of the structure, the larger its perturbation effects.
Unfortunately, this is not that evident to classify the problems that will behave bad with re- spect to a reduced basis approach (such as the cyclindrical sample) from those which will be- have well. Indeed, the other test-cases presented in this typescript (namely the rectangular ten- sion sample of section 6.2and the Tada’s crack of section 7.1) behave well even with a ratio crack size/structure sizecomparable or higher than the one of the cylindrical sample. To illus- trate that point, a similar analysis of the solution displacement fields has been conducted for those test-cases. Hence, on Figure2.33, the initial displacement field presented on the initial config- uration, as well as the final field on the final configuration are plotted, for both the rectangular tension sample and the Tada’s crack. The fields of the rectangular tension sample are rather diffi- cult to grasp because they are pointing toward a node fixed to avoid rigid body motion during the computations. However, as a matter of fact, the crack become large, but the solution fields do not evolve very much during the propagation. Considering the Tada’s crack field evolutions, again it is very clear that all along the propagation, the obtained displacement field remain very similar.
To conclude on that point, the statement on the good or bad behavior of a specific test-case with respect to the reduced basis approach should be qualified because the proposed reduced basis method exhibit some convergence properties. In other words, it always exist asmall enough propagation range in which the method will be efficient. In addition, the extra cost brought by the use of the proposed model reduction technique on problems for which it is not efficient is marginal. In this respect, attempts of applying it on any type of crack propagation computation can always been envisaged. Finally, indicators allowing to estimatea priori the quality of the behavior of a test-case with respect to a reduced basis approach would be really handy, and it could be an interesting perspective for future work.