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We present a first-order method for solving constrained optimization problems. The method is derived from our previous work, a modified search direction method inspired by singular value decomposition. In this work, we simplify its computational framework to a ``gradient descent akin'' method (GDAM), i.e., the search direction is computed using a linear combination of the negative and normalized objective and constraint gradient. We give fundamental theoretical guarantees on the global convergence of the method. This work focuses on the algorithms and applications of GDAM. We present computational algorithms that adapt common strategies for the gradient descent method. We demonstrate the potential of the method using two engineering applications, shape optimization and sensor network localization. When practically implemented, GDAM is robust and very competitive in solving the considered large and challenging optimization problems.

This work studies shape filtering techniques, namely the convolution-based (explicit) and the PDE-based (implicit), and introduces an implicit bulk-surface filtering method to control the boundary smoothness and preserve the internal mesh quality simultaneously in the course of bulk (solid) shape optimization. To that end, volumetric mesh is filtered by the solution of pseudo-solid governing equations which are stiffened by the mesh-Jacobian and endowed with the Robin boundary condition which involves the Laplace-Beltrami operator on the mesh boundaries. Its superior performance from the non-simultaneous (sequential) treatment of boundary and internal meshes is demonstrated for the shape optimization of a complex solid structure. Well-established explicit filters, namely Gaussian and linear, and the Helmholtz/Sobolev-based (implicit) filter are critically examined in terms of consistency (rigid-body-movement production), geometric characteristics and the computational cost. It is shown that the implicit filtering is numerically more efficient and unconditionally consistent, compared to the explicit one. Supported by numerical experiments, a regularized Green's function is introduced as an equivalent explicit form of the Helmholtz/Sobolev filter. Furthermore, we give special attention to derive mesh-independent filtered sensitivities for node-based shape optimization with non-uniform meshes. It is shown that the mesh independent filtering can be achieved by scaling discrete sensitivities with the inverse of the mesh mass matrix.

We introduce a novel hybrid methodology combining classical finite element methods (FEM) with neural networks to create a well-performing and generalizable surrogate model for forward and inverse problems. The residual from finite element methods and custom loss functions from neural networks are merged to form the algorithm. The Finite Element Method-enhanced Neural Network hybrid model (FEM-NN hybrid) is data-efficient and physics conforming. The proposed methodology can be used for surrogate models in real-time simulation, uncertainty quantification, and optimization in the case of forward problems. It can be used for updating the models in the case of inverse problems. The method is demonstrated with examples, and the accuracy of the results and performance is compared against the conventional way of network training and the classical finite element method. An application of the forward-solving algorithm is demonstrated for the uncertainty quantification of wind effects on a high-rise buildings. The inverse algorithm is demonstrated in the speed-dependent bearing coefficient identification of fluid bearings. The hybrid methodology of this kind will serve as a paradigm shift in the simulation methods currently used.

Reducing the intensity of wind excitation via aerodynamic shape modification is a major strategy to mitigate the reaction forces on supertall buildings, reduce construction and maintenance costs, and improve the comfort of future occupants. To this end, computational fluid dynamics (CFD) combined with state-of-the-art stochastic optimization algorithms is more promising than the trial and error approach adopted by the industry. The present study proposes and investigates a novel approach to risk-averse shape optimization of tall building structures that incorporates site-specific uncertainties in the wind velocity, terrain conditions, and wind flow direction. A body-fitted finite element approximation is used for the CFD with different wind directions incorporated by re-meshing the fluid domain. The bending moment at the base of the building is minimized, resulting in a building with reduced cost, material, and hence, a reduced carbon footprint. Both risk-neutral and risk-averse optimization of the twist and tapering of a representative building are presented under uncertain inflow wind conditions that have been calibrated to fit freely-available site-specific data from Basel, Switzerland. The risk-averse strategy uses the conditional value-at-risk to optimize for the low-probability high-consequence events appearing in the worst 10% of loading conditions. Adaptive sampling is used to accelerate the gradient-based stochastic optimization pipeline. The adaptive method is easy to implement and particularly helpful for compute-intensive simulations because the number of gradient samples grows only as the optimal design algorithm converges. The performance of the final risk-averse building geometry is exceptionally favorable when compared to the risk-neutral optimized geometry, thus, demonstrating the effectiveness of the risk-averse design approach in computational wind engineering.

This work presents a partitioned solution procedure to compute shape gradients in fluid-structure interaction (FSI) using black-box adjoint solvers. Special attention is paid to project the gradients onto the undeformed configuration. This is due to the mixed Lagrangian-Eulerian formulation of large-displacement FSI in this work. Adjoint FSI problem is partitioned as an assembly of well-known adjoint fluid and structural problems, without requiring expensive cross-derivatives. The sub-adjoint problems are coupled with each other by augmenting the target functions with auxiliary functions, independent of the concrete choice of the underlying adjoint formulations. The auxiliary functions are linear force-based or displacement-based functionals which are readily available in well-established single-disciplinary adjoint solvers. Adjoint structural displacements, adjoint fluid displacements, and domain-based adjoint sensitivities of the fluid are the coupling fields to be exchanged between the adjoint solvers. A reduced formulation is also derived for the case of boundary-based adjoint shape sensitivity analysis for fluids. Numerical studies show that the complete formulation computes accurate shape gradients whereas inaccuracies appear in the reduced gradients, specially in regions of strong flow gradients and near singularities. Nevertheless, reduced gradient formulations are found to be a compromise between computational costs and accuracy. Mapping techniques including nearest element interpolation and the mortar method are studied in computational adjoint FSI. It is numerically shown that the mortar method does not introduce spurious oscillations in primal and sensitivity fields along non-matching interfaces, unlike the nearest element interpolation.

We propose a first-order method for solving inequality constrained optimization problems. The method is derived from our previous work [12], a modified search direction method (MSDM) that applies the singular-value decomposition of normalized gradients. In this work, we simplify its computational framework to a "gradient descent akin" method, i.e., the search direction is computed using a linear combination of the negative and normalized objective and constraint gradient. The main focus of this work is to provide a mathematical aspect to the method. We analyze the global behavior and convergence of the method using a dynamical systems approach. We then prove that the resulting trajectories find local solutions by asymptotically converging to the central path(s) for the logarithmic barrier interior-point method under the so-called relative convex condition. Numerical examples are reported, which include both common test examples and applications in shape optimization., Comment: 39 pages, 9 figures