by Trond Kvamsdal
NTNU Norwegian University of Science and Technology
Background
Reduced-basis methods (RBMs) offer an efficient framework for computing real-time approximations of parameterized PDEs. Their main limitation is the requirement of parametric affinity, which is rarely satisfied in practical applications and is essential for rapid online assembly. A common remedy is to introduce approximate parametric affinity, but existing approaches are often highly intrusive, demanding substantial expert knowledge of both the problem and the high-fidelity solver, and frequently requiring source-code modifications. In [1] we propose a minimally intrusive approach to approximate affinization based on least-squares projections onto a prescribed function space.
Least square projection method for affinization
An affine (or parametrically affine) problem is one in which all parameter dependence can be expressed as a finite sum of parameter-dependent scalars multiplied by parameter-independent operators (see [2], Figure 1.1). The importance of this property lies in the fact that, for affine problems, all reduced-order vectors and matrices can be precomputed a priori during the so-called offline stage (see [2], Figure 1.1).
Classical techniques for handling non-affine problems include the Empirical Interpolation Method (EIM) and its discrete variant, DEIM. However, instead of interpolating or modifying the high-fidelity solver as in EIM/DEIM, in [1] we construct an approximate affine decomposition of the reduced operators using a least-squares projection. This approach builds the approximation directly from a least-squares projection of the reduced quantities, avoiding the need for deep access to the high-fidelity code, which is required by EIM and DEIM.
Numerical example
In [1], we first demonstrate that our approach performs very well for shape optimization in linear elasticity (see [1], Section 5.2). Here, we present results obtained for a jacket foundation structure of a bottom-fixed offshore wind turbine. The model was developed using Sesam GeniE.
We replace eight of the tubular beam-element joints with three-dimensional shell-element reduced-basis components (Figure 15). Each component has six interfaces, each with three translational and three rotational degrees of freedom. The degrees of freedom on the circular boundaries of the shell elements are condensed to a single point using explicit constraint equations.
The finite-element discretization of each component consists of 26,241 first-order linear shell elements, including both triangular (three-noded) and quadrilateral (four-noded) elements. The resulting number of degrees of freedom is 𝑁ℎ = 153, 132.
Each component is parameterized by the wall thickness 𝜇 of the top and bottom braces. We consider 𝜇 ∈ = [10−2, 10−1] m. The parameter enters the equations through the thickness in the shell finite-element stiffness matrices. However, thanks to the automatic affinization procedure, detailed knowledge of the parameterization is not required.
For the discretization of the remaining jacket structure, we use 102 first order (two-noded) beam elements, with a single beam element connecting each pair of joints. The tower is modelled using multiple beam elements of varying diameters to approximate a conical shape. This part of the structure is not parameterized.
The results show that using a polynomial order p=3 in the least-squares projection, together with four reduced-order basis modes, achieves a relative energy error of less than 1%, while providing a significant speedup compared to static condensation—the most effective alternative (see [1]).
| Figur 1a) The OWT jacket foundation and tower structure with the load 𝑓applied at the tower-turbine interface node and acts on translational and rotational degrees of freedom. | Figure 1b) Three-dimensional shell-element joint component with six interfaces (left). The component is parameterized by the identical wall thickness 𝜇 of the four brace cylinders. | Figure 1c) The finite-element discretization of each component consists of 𝑁ℎ = 153 132 number of degrees of freedom. |
| Figure 2. Example of solutions shown in Sesam visualization software (Xtract). Displacements with the field reconstructed in two of the eight joint components (left); Von-Mises stresses on one of the reconstructed joint components (right). | |
Acknowledgement
We are grateful for the support received by the Research Council of Norway and the industrial partners of the RaPiD project (DNV and Dr. Techn. Olav Olsen) under grant number 313909.
References
[1] E. Fonn, H. v. Brummelen, J. L. Eftang, T.Rusten, K. A. Johannessen, T. Kvamsdal, A. Rasheed: Least-Squares Projected Models for Non-Intrusive Affinization of Reduced Basis Methods. International Journal for Numerical Methods in Engineering, 2025; 126:e70127, https://doi.org/10.1002/nme.70127
[2] A. Quarteroni, A. Manzoni, and F. Negri: Reduced Basis Methods for Partial Differential Equations: An Introduction. UNITEXT, Springer International Publishing, 2016.