Abstract
This work introduces and compares two different CAD-based mesh deformation methods. The methods are used within an adjoint structural shape optimization, which is part of an evolving CAD-based adjoint multidisciplinary optimization framework for turbomachinery components. During an optimization, the CAD geometry is updated at each design iteration, such that the structural mesh has to be deformed appropriately. The mesh is deformed in three stages. First, the nodes along the edges of the outer mesh are displaced to match the shape of the CAD edges, which are given by B-spline curves. Next, the remaining outer mesh nodes are displaced to match the shape of the CAD faces, which are given by B-spline surfaces. Finally, the outer mesh node deformations are used to solve for the inner node deformations using either an inverse distance interpolation or the linear elasticity analogy. Coupling the mesh deformation with an adjoint structural solver enables gradient computations of structural constraints with respect to CAD design parameters. To compare the robustness of the two mesh deformation methods, a CAD-based structural shape optimization using each method was performed.
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Abbreviations
- \(m \in \mathbb {N}\) :
-
number of FEM mesh nodes
- \(m_i \in \mathbb {N}\) :
-
number of inner FEM mesh nodes
- \(m_o \in \mathbb {N}\) :
-
number of outer FEM mesh nodes
- \(n \in \mathbb {N}\) :
-
number of CAD design parameters
- \(u, v \in \mathbb {R}\) :
-
B-Spline foot points
- \(u_B \in \mathbb {R}\) :
-
foot point of begin vertex
- \(u_B^M \in \mathbb {R}\) :
-
morphed foot point of begin vertex
- \(u_E \in \mathbb {R}\) :
-
foot point of end vertex
- \(u_E^M \in \mathbb {R}\) :
-
morphed foot point of end vertex
- \(\mathbf {b} \in \mathbb {R}^{3m}\) :
-
load vector
- \(\mathbf {u} \in \mathbb {R}^{3m}\) :
-
FEM mesh displacements
- \(\mathbf {u}_{inner} \in \mathbb {R}^{3m_i}\) :
-
inner FEM mesh displacements
- \(\mathbf {u}_{outer} \in \mathbb {R}^{3m_o}\) :
-
outer FEM mesh displacements
- \(\mathbf {x} \in \mathbb {R}^{3m}\) :
-
FEM mesh coordinates
- \(\bar{\mathbf {x}} \in \mathbb {R}^{3m}\) :
-
adjoint FEM mesh coordinates
- \(A \in \mathbb {R}^{3m \times 3m}\) :
-
stiffness matrix
- \(C \in \mathbb {R}^3\) :
-
B-spline curve
- \(C^M \in \mathbb {R}^3\) :
-
morphed B-spline curve
- \(E \in \mathbb {R}\) :
-
Young’s modulus
- \(P \in \mathbb {R}^3\) :
-
mesh point
- \(P^M \in \mathbb {R}^3\) :
-
morphed mesh point
- \(S \in \mathbb {R}^3\) :
-
B-spline surface
- \(S^M \in \mathbb {R}^3\) :
-
morphed B-spline surface
- \(V_B \in \mathbb {R}^3\) :
-
begin vertex
- \(V_B^M \in \mathbb {R}^3\) :
-
morphed begin vertex
- \(V_E \in \mathbb {R}^3\) :
-
end vertex
- \(V_E^M \in \mathbb {R}^3\) :
-
morphed end vertex
- \(\nu \in \mathbb {R}\) :
-
Poisson’s ratio
- \(\sigma _{max} \in \mathbb {R}\) :
-
maximum von Mises stress
- \(\bar{\sigma }_{max} \in \mathbb {R}\) :
-
adjoint maximum von Mises stress
- \(\varvec{\alpha } \in \mathbb {R}^n\) :
-
CAD design parameters
- \(\bar{\varvec{\alpha }} \in \mathbb {R}^n\) :
-
adjoint CAD design parameters
- \(\Delta \in \mathbb {R}\) :
-
steepest descent step size
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Acknowledgements
The work presented in this paper has received funding from the European Commission through the IODA project under grant agreement number 642959.
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Schwalbach, M., Verstraete, T., Müller, JD., Gauger, N. (2019). A Comparative Study of Two Different CAD-Based Mesh Deformation Methods for Structural Shape Optimization. In: Andrés-Pérez, E., González, L., Periaux, J., Gauger, N., Quagliarella, D., Giannakoglou, K. (eds) Evolutionary and Deterministic Methods for Design Optimization and Control With Applications to Industrial and Societal Problems. Computational Methods in Applied Sciences, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-89890-2_4
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