Abstract
In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; an a posteriori error estimation procedures—rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor; and Offline-Online computational decomposition strategies—minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities—to model the materials and loads—and geometrical parameters—to model different geometrical configurations—with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.
Keywords
- RB Approximation
- Posteriori Error Estimators
- Basis Reduction (RB)
- Galerkin Projection
- Central Crack Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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- 1.
For the sake of simple illustration, we omit the “original” superscript o on (r, z, θ).
- 2.
- 3.
Here we note that, the Young’s modulus E in the isotropic and axisymmetric cases or E 1, E 2 and E 3 in the orthotropic case only.
- 4.
The average tangential displacement on \(\varGamma ^{\mathrm {o}}_N\) is not exactly s(μ) but rather \(s({\boldsymbol {\mu }})/l_{\varGamma ^{\mathrm {o}}_N}\), where \(l_{\varGamma ^{\mathrm {o}}_N}\) is the length of \({\varGamma ^{\mathrm {o}}_N}\). It is obviously that the two descriptions of the two outputs, “integral of” and “average of”, are pretty much equivalent to each other.
- 5.
Note that since Dirichlet boundary nodes are eliminated from the FE system.
- 6.
It is possible to show that the bifurcation point is related to the eigenvalue of the linearized model [5], so we are able to set in a proper way the range of the parameter domain.
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Acknowledgements
We are sincerely grateful to Prof. A.T. Patera (MIT) and Dr. C.N. Nguyen (MIT) for important suggestions, remarks, insights, and codevelopers of the rbMIT and RBniCS (http://mathlab.sissa.it/rbnics) software libraries used for the numerical tests. We acknowledge the European Research Council consolidator grant H2020 ERC CoG 2015 AROMA-CFD GA 681447 (PI Prof. G. Rozza).
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Appendix
Appendix
8.1.1 Stress-Strain Matrices
In this section, we denote E i, i = 1, 3 as the Young’s moduli, ν ij; i, j = 1, 2, 3 as the Poisson ratios; and G 12 as the shear modulus of the material.
8.1.1.1 Isotropic Cases
For both of the following cases, E = E 1 = E 2, and ν = ν 12 = ν 21.
Isotropic plane stress:
Isotropic plane strain:
8.1.1.2 Orthotropic Cases
Here we assume that the orthotropic material axes are aligned with the axes used for the analysis of the structure. If the structural axes are not aligned with the orthotropic material axes, orthotropic material rotation must be rotated by with respect to the structural axes. Assuming the angle between the orthogonal material axes and the structural axes is θ, the stress-strain matrix is given by \([{\mathbf {E}}] = [{{\boldsymbol {T}}}(\theta )][\hat {{\mathbf {E}}}][{{\boldsymbol {T}}}(\theta )]^T\), where
Orthotropic plane stress:
Note here that the condition
must be required in order to yield a symmetric [E].
Orthotropic plane strain:
Here Λ = (1 − ν 13 ν 31)(1 − ν 23 ν 32) − (ν 12 + ν 13 ν 32)(ν 21 + ν 23 ν 31). Furthermore, the following conditions,
must be satisfied, which leads to a symmetric [E].
An reasonable good approximation for the shear modulus G 12 in orthotropic case is given by [10] as
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Huynh, D.B.P., Pichi, F., Rozza, G. (2018). Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings. In: Di Pietro, D., Ern, A., Formaggia, L. (eds) Numerical Methods for PDEs. SEMA SIMAI Springer Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-94676-4_8
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