Gradient-based Optimization Framework Research Articles

Online Trajectory Replanning for Avoiding Moving Obstacles Using Fusion Prediction and Gradient-Based Optimization

In this study, we introduce a novel method for an online trajectory replanning approach for fixed-wing Unmanned Aerial Vehicles (UAVs). Our method integrates moving obstacle predictions within a gradient-based optimization framework. The trajectory is represented by uniformly discretized waypoints, which serve as the optimization variables within the cost function. This cost function incorporates multiple objectives, including obstacle avoidance, kinematic and dynamic feasibility, similarity to the reference trajectory, and trajectory smoothness. To enhance prediction accuracy, we combine physics-based and pattern-based methods for predicting obstacle movements. These predicted movements are then integrated into the online trajectory replanning framework, significantly enhancing the system’s safety. Our approach provides a robust solution for navigating dynamic environments, ensuring both optimal and secure UAV operation.

Shape Optimization for Lithium-Ion Battery with Porous Electrodes

In this presentation, we introduce a gradient-based shape optimization framework to design porous electrodes for maximum energy storage. Lithium-ion batteries are becoming an increasingly important energy source and storage device. Large surface area that advocates high reaction rates often leads to small porosities that deteriorate electrochemical transport, which significantly restricts ion penetration depth, limiting the material utilization in planar electrodes. Consequently, architected electrodes are required to optimize performance.We consider a model with two materials, namely porous electrode and pure electrolyte. Density-based topology optimization with continuous material volume fraction has previously been implemented on lithium-ion battery system for both half-cell and full-cell model to maximizes its energy storage. Despite the capability of producing complex interdigitated structures, an appropriate penalization scheme for intermediate materials is required for the optimizer to provide near binary designs. Setting up such a penalization is often time-consuming due to the large number of design-dependent variables that affect the forward model differently. Alternatively, the shape optimization approach used in this work produces binary designs naturally by morphing conformal meshes, circumventing inaccuracies associated with fuzzy interfaces. Shape sensitivities are computed via the adjoint method and leveraging automatic differentiation. The optimization problem is solved using a nonlinear programming technique.In this presentation, we perform shape optimization on a half-cell model to maximize its energy storage when a fixed charging current is applied. We study multiple arrangements of physical parameters and compare their performance against a monolithic electrode. Both 2D and 3D results are presented.

Topology Optimization of Hard-Magnetic Soft Phononic Structures for Wide Magnetically Tunable Band Gaps

Abstract Hard-magnetic soft materials, which exhibit finite deformation under magnetic loading, have emerged as a promising class of soft active materials for the development of phononic structures with tunable elastic wave band gap characteristics. In this paper, we present a gradient-based topology optimization framework for designing the hard-magnetic soft materials-based two-phase phononic structures with wide and magnetically tunable anti-plane shear wave band gaps. The incompressible Gent hyperelastic material model, along with the ideal hard-magnetic soft material model, is used to characterize the constitutive behavior of the hard-magnetic soft phononic structure phases. To extract the dispersion curves, an in-house finite element model in conjunction with Bloch’s theorem is employed. The method of moving asymptotes is used to iteratively update the design variables and obtain the optimal distribution of the hard-magnetic soft phases within the phononic structure unit cell. Analytical sensitivity analysis is performed to evaluate the gradient of the band gap maximization function with respect to each one of the design variables. Numerical results show that the optimized phononic structures exhibit a wide band gap width in comparison to a standard hard-magnetic soft phononic structure with a central circular inclusion, demonstrating the effectiveness of the proposed numerical framework. The numerical framework presented in this study, along with the derived conclusions, can serve as a valuable guide for the design and development of futuristic tunable wave manipulators.

A sensitivity-based estimation method for investigating control co-design relevance

Abstract. Control co-design is a promising approach for wind turbine design due to the importance of the controller in power production, stability, load alleviation, and the resulting coupled effects on the sizing of the turbine components. However, the high computational effort required to solve optimization problems with added control design variables is a major obstacle to quantifying the benefit of this approach. In this work, we propose a methodology to identify if a design problem can benefit from control co-design. The estimation method, based on post-optimum sensitivity analysis, quantifies how the optimal objective value varies with a change in control tuning. The performance of the method is evaluated on a tower design optimization problem, where fatigue load constraints are a major driver, and using a linear quadratic regulator targeting fatigue load alleviation. We use the gradient-based multi-disciplinary optimization framework Cp-max. Fatigue damage is evaluated with time-domain simulations corresponding to the certification standards. The estimation method applied to the optimal tower mass and optimal cost of energy show good agreement with the results of the control co-design optimization while using only a fraction of the computational effort. Our results additionally show that there may be little benefit to using control co-design in the presence of an active frequency constraint. However, for a soft–soft tower configuration where the resonance can be avoided with active control, using control co-design results in a taller tower with reduced mass.

Causality-aware social recommender system with network homophily informed multi-treatment confounders

Typical recommender systems utilize observed ratings of users as inputs to learn their preferences and aim to output recommendations of new items that users will like by predicting their potential ratings. The real world is driven by causality given that whether or not a user is made exposed to the item may significantly affect the rating. While some recent methods have taken a causal view to mitigate the confounding bias in observed rating data, few have recognized recommendation as a multiple causal inference problem that concerns numerous items simultaneously in practice. By framing the recommendation as a multiple causal inference problem, this paper develops a causality-aware social recommender that incorporates underlying social network structures with matrix factorization methods for deconfounding in networked observational data to enhance social recommendations. Leveraging social network information, which inherently confounds with the preferences of users, is particularly crucial but not trivial for inferring unobserved confounders in social recommender systems. Considering that connected users in a social network share similar attributes, we propose to incorporate the network homophily into the matrix factorization models through regularization to better learn the latent variables. The extracted latent variables then capture the network informed multi-treatment confounders to mitigate the confounding bias and improve the rating prediction accuracy. The models are estimated through a proximal gradient-based optimization framework, which not only eases the incorporation of the network structure constraints but also improves the computational efficiency of the proposed method. Simulation and case studies are conducted to validate the proposed method.

A duality-preserving adjoint method for segregated Navier–Stokes solvers

Adjoint methods efficiently compute gradients for systems with many inputs and have been widely used for large-scale gradient-based optimization in fluid mechanics. To ensure optimization's numerical robustness, we need to develop an adjoint solution algorithm that has a similar, if not the same, convergence rate as the primal flow solver at each optimization iteration. This consistent primal-adjoint convergence behavior is also called duality-preserving (DP). Existing DP adjoint methods are mostly for fully coupled Navier–Stokes (NS) solvers with compressible flows. However, few DP adjoints have been proposed for segregated NS solvers, e.g., the semi-implicit method for pressure-linked equations (SIMPLE) algorithm solvers widely used for incompressible flow simulations. This study will fill this gap by deriving a fixed-point segregated adjoint formulation that fully preserves the convergence behavior of primal solvers. We first rewrite the steady-state segregated NS primal solution process into a fully coupled left-preconditioned Richardson format. Then, we transpose the preconditioner matrix to obtain a DP iterative adjoint formulation. In addition to algebraic derivation, we create a new graph representation that significantly streamlines the DP adjoint development for new segregated solvers. We prove that the proposed graph representation is equivalent to the above preconditioner transpose approach and guarantees duality. We evaluate our proposed algorithm using three cases with increasing complexity: a miniature-sized problem, an airfoil, and a wing. To quantify duality, we compare the eigenvalues between the primal and adjoint solvers and observe excellent agreements. We also evaluate to what extent various numerical settings (e.g., inner iteration tolerances and non-dual preconditioner) impact the eigenvalue distributions. Our proposed adjoint achieves machine-precision accurate gradient computation with competitive speed. Finally, we incorporate our adjoint solver into a large-scale gradient-based optimization framework and demonstrate its capability for wing aerodynamic shape optimization. The proposed fixed-point adjoint approach has the potential to make the adjoint solution more robust and efficient for any segregated NS solvers.

Construction aware optimization of concrete plate thicknesses

Concrete is a significant contributor to the global CO2 emissions, and therefore effort is being made to reduce its consumption in buildings and infrastructure. Since concrete flat slabs constitute most of the concrete in buildings, reducing their weight has great potential to reduce the overall environmental burden caused by buildings. In this study the authors develop an efficient, gradient-based thickness optimization framework for minimizing the concrete volume in concrete slabs, while insuring feasibility and constractability of the design. The results show that concrete savings of up to 35% are possible without increasing the structural depth, where in most cases minor increase in construction complexity is sufficient for significant concrete savings. Therefore, an appealing trade-off between the concrete savings, construction complexity, and architectural cost is revealed and can be used by engineers to reduce concrete consumption in buildings. The optimized thickness designs are not trivial and represent a balance between competing requirements that is obtained by the optimization algorithm, therefore the results demonstrate the potential of structural optimization as a design tool in structural engineering.

Gradient-based concurrent topology and anisotropy optimization for mechanical structures

The present paper introduces a novel gradient-based optimization framework to obtain discrete topologies consisting of either none or complete presence of anisotropic material, modeled by means of the polar formalism. Current topology optimizations with the polar formalism are based on optimality criteria, and limited to performing compliance minimization for thermodynamically feasible materials. The proposed optimization approach uses a sequential approximations approach, based on the Methods of Moving Asymptotes. The material density, orientation and anisotropic modules are updated separately at each iteration, in parallel sub-problems constructed with different types of approximations and settings. The proposed optimization approach is successfully validated for compliance minimization against the Alternate Directions method for general orthotropic materials, defined by the thermodynamic bounds. The importance of the anisotropy initialization in the gradient-based approach is highlighted to obtain stiffer solutions. The gradient-based strategy is also extended to incorporate geometric bounds on the polar parameters, defining the domain of existence of composite laminates. Obtained results for compliance minimization with laminates are compared to published results using lamination parameters. Finally, new solutions are presented showing the improvement of the compliance with the expansion of the anisotropy design domains. This paper paves the way for strength-based topology optimization of parts made of orthotropic materials or composite laminates.

Simultaneous wing shape and actuator parameter optimization using the adjoint method

Increasing attention to urban air mobility with distributed electric propulsion causes an increased interest in propeller aircraft design and optimization. Various studies have focused on designing and optimizing the aerodynamic performance of wings and propellers. However, existing studies optimized the wing and propeller separately due to the numerical challenges in simulations and optimization. It is not clear to what extent high-fidelity coupled wing-propeller aerodynamic optimization can benefit the propeller aircraft design. As a first step to answer the above question, this study develops the capability to simultaneously optimize the wing shape and actuator parameters. We use a high-fidelity computational fluid dynamics solver to simulate the wing aerodynamics, and the propeller is modeled as an actuator disk. We develop a smoothed actuator disk formulation that allows us to change the actuator's location and radius during the optimization, along with the wing shape. We use the discrete adjoint approach to compute the derivatives and couple them with a gradient-based optimization framework. The benefit of the above adjoint-based optimization framework is that it can use a large number of design variables to allow large design freedom for both wing and actuator. We perform 15 optimizations to evaluate how various actuator parameters impact the wing aerodynamic performance; both single and double actuator configurations are considered. We observe that the spanwise location and outer radius are the two most important actuator parameters, and using more actuator parameters as the design variables result in better performance. For example, optimizing the wing shape while allowing the actuator location and outer radius to move exhibits 11.4% more drag reduction compared with optimizing the wing shape while fixing the actuator parameters. This study serves as the starting point for more detailed high-fidelity coupled wing-propeller aerodynamic optimizations.

Optimization Over Time-Varying Networks and Unbounded Information Delays

Solving optimization problems in multiagent systems involves information exchange between agents. The obtained solutions should be robust to information delays and errors that arise from an unreliable wireless network, which typically connects the multiagent system. In today’s large-scale dynamic Internet of Things style multiagent scenarios, the network topology changes and evolves over time. In this article, we present a simple distributed gradient-based optimization framework and an associated algorithm. Convergence to a minimum of a given objective is shown under mild conditions on the network topology and objective. A key feature of our approach is that we merely assume that the messages sent reach the intended receiver, possibly delayed, with some positive probability. To the best of authors’ knowledge, ours is the first analysis under such weak general network conditions. We also discuss in detail the verifiability of the assumptions involved. This article also makes a technical contribution in terms of the allowed class of objective functions. Specifically, we present an analysis wherein the objective function is such that its sample-gradient is merely locally Lipschitz continuous. The theory developed herein is supported by empirical results.

Efficient strength optimization of variable stiffness laminates using lamination parameters with global failure index

A computationally-efficient strength optimization method tailoring novel composite laminates using lamination parameters is developed. The method adopts a global p-norm approach to aggregate local failure indices into a global failure index, based on the Tsai-Wu failure criterion. For design purposes, the novel composite laminates are characterized via lamination parameters that can subsequently be transformed into locally variable fiber orientations in an existing three-step optimization method. An elliptical formulation of the conservative failure envelope is applied to represent the Tsai-Wu criterion in terms of lamination parameters. A lamination-parameter-based two-level approximation for the global failure index is derived, which guarantees the anticipated conservativeness and convexity in a gradient-based optimization framework. Numerical results show that the computational efficiency of the proposed strength optimization method improves remarkably with a proper value of p, compared to the existing local-based min-max method. The method is also shown to be robust and generate converged optimum designs even in the presence of stress concentrations and singularities.

Breaking the Convergence Barrier: Optimization via Fixed-Time Convergent Flows

Accelerated gradient methods are the cornerstones of large-scale, data-driven optimization problems that arise naturally in machine learning and other fields concerning data analysis. We introduce a gradient-based optimization framework for achieving acceleration, based on the recently introduced notion of fixed-time stability of dynamical systems. The method presents itself as a generalization of simple gradient-based methods suitably scaled to achieve convergence to the optimizer in a fixed-time, independent of the initialization. We achieve this by first leveraging a continuous-time framework for designing fixed-time stable dynamical systems, and later providing a consistent discretization strategy, such that the equivalent discrete-time algorithm tracks the optimizer in a practically fixed number of iterations. We also provide a theoretical analysis of the convergence behavior of the proposed gradient flows, and their robustness to additive disturbances for a range of functions obeying strong convexity, strict convexity, and possibly nonconvexity but satisfying the Polyak-Łojasiewicz inequality. We also show that the regret bound on the convergence rate is constant by virtue of the fixed-time convergence. The hyperparameters have intuitive interpretations and can be tuned to fit the requirements on the desired convergence rates. We validate the accelerated convergence properties of the proposed schemes on a range of numerical examples against the state-of-the-art optimization algorithms. Our work provides insights on developing novel optimization algorithms via discretization of continuous-time flows.

Comparison of aerodynamic planform optimization of non-planar rotors using blade element momentum method and a vortex cylinder model

The present work compares non-planar rotors designed using the blade element momentum (BEM) method and a vortex cylinder model. In a previous work, it is shown that blade element theory coupled with the superposition of the vortex cylinder model (BEVC) is able to model the loads of non-planar rotors. The result predicted by the BEVC model is in significantly improved agreement with higher-fidelity models than the loads as predicted using the BEM method. In this work, the BEM method and the BEVC method are integrated into a gradient-based optimization framework for aerodynamic planform optimization, in which the analytical gradients are obtained using the algorithmic differentiation (AD) method. In the present study, the rotor is assumed to be stiff for all cases such that the pure aerodynamic effects are highlighted. Loads of the optimized non-planar rotors with different geometries under different constraints designed from both methods are calculated using the BEM method, the BEVC method and also the higher-fidelity lifting-line (LL) method. Within the constraints of the present work it was found that the advantage of the BEVC method is not significant when comparing the integrated aerodynamic loads: the non-planar rotor designed using the BEM method gives similar total thrust and power as the rotor designed using the BEVC method when the designs are evaluated with the higher-fidelity LL method. However, the results confirmed that the distributed aerodynamic loads of the non-planar rotors predicted by the BEVC method are in improved agreement with the LL method compared to the BEM method.

Multirow Adjoint-Based Optimization of NICFD Turbomachinery Using a Computer-Aided Design-Based Parametrization

Abstract Currently, most of the adjoint-based design systems documented in the open literature assume that the fluid behaves as an ideal gas, are restricted to the optimization of a single row of blades, or are not suited to impose geometric constraints. In response to these limitations, this paper presents a gradient-based shape optimization framework for the aerodynamic design of turbomachinery blades operating under nonideal thermodynamic conditions. The proposed design system supports the optimization of multiple blade rows, and it integrates a computer-aided design (CAD)-based parametrization with a Reynolds-averaged Navier–Stokes (RANS) flow solver and its discrete adjoint counterpart. The capabilities of the method were demonstrated by performing the design optimization of a single-stage axial turbine that employs isobutane (R600a) as working fluid. Notably, the aerodynamic optimization respected the minimum thickness constraint at the trailing edge of the stator and rotor blades and reduced the entropy generation within the turbine by 36%, relative to the baseline, which corresponds to a total-to-total isentropic efficiency increase of about 4 percentage points. The analysis of the flow field revealed that the performance improvement was achieved due to the reduction of the wake intensity downstream of the blades and the elimination of a shock-induced separation bubble at the suction side of the stator cascade.

Smart Gradient - An adaptive technique for improving gradient estimation

<p style='text-indent:20px;'>Computing the gradient of a function provides fundamental information about its behavior. This information is essential for several applications and algorithms across various fields. One common application that requires gradients are optimization techniques such as stochastic gradient descent, Newton's method and trust region methods. However, these methods usually require a numerical computation of the gradient at every iteration of the method which is prone to numerical errors. We propose a simple limited-memory technique for improving the accuracy of a numerically computed gradient in this gradient-based optimization framework by exploiting (1) a coordinate transformation of the gradient and (2) the history of previously taken descent directions. The method is verified empirically by extensive experimentation on both test functions and on real data applications. The proposed method is implemented in the <inline-formula><tex-math id="M1">\begin{document}$\texttt{R} $\end{document}</tex-math></inline-formula> package <inline-formula><tex-math id="M2">\begin{document}$ \texttt{smartGrad}$\end{document}</tex-math></inline-formula> and in C<inline-formula><tex-math id="M3">\begin{document}$ \texttt{++} $\end{document}</tex-math></inline-formula>.</p>

Structural Analysis and Optimization of Wings Subjected to Dynamic Loads

Dynamic loads arising from the Title 14 Code of Federal Regulations (14 CFR; “Aeronautics and Space,” Federal Aviation Administration, U.S. Dept. of Transportation, Subchapter C Aircraft, Part 25 Airworthiness Standards: Transport Category Airplanes, https://ecfr.federalregister.gov/current/title-14/chapter-I/subchapter-C/part-25 [retrieved 23 August 2021]) are critical drivers of aircraft design. The aircraft structural design process is an iterative one involving computationally expensive simulations. Existing methods simplify the dynamic loads to equivalent static ones and design the structure using shell-based analysis. Other literature approaches use beam models for structural sizing but make assumptions on the geometry, material distribution, or both. In this work, the use of the Variational Asymptotic Method is explored to 1) systematically reduce the three-dimensional (3-D) wingbox to a one-dimensional (1-D) model, 2) solve the 1-D model using a nonlinear beam theory, and 3) recover the stress field on the wingbox. An adjoint-driven gradient-based structural optimization framework is developed to size the structure for dynamic loads subjected to strength-based failure considerations. The proposed structural analysis method is applied to the wingbox structural analysis and sizing of a novel hybrid-electric propulsion aircraft: NASA’s Parallel Electric-Gas Architecture with Synergistic Utilization Scheme (PEGASUS) concept. Results in the paper show that the sizing of aircraft wings for 14 CFR specified maneuvers 2.5 and static maneuvers using the proposed approach produces a 6% difference in weight compared to a shell-based method but with a speedup of 7.8 times, and it is efficient at sizing the wingbox structure for dynamic loads generated by gust maneuvers.

3-D high-fidelity hydrostructural optimization of cavitation-free composite lifting surfaces

Design and manufacturing technologies have promoted the use of composites in hydrodynamic lifting surfaces. Composites, even with state-of-the-art coatings, are susceptible to cavitation erosion damage. This work aims to design a cavitation-free composite lifting surface to maximize efficiency while ensuring structural integrity. We optimize a canonical hydrofoil using a gradient-based optimization framework that couples a Reynolds-averaged Navier–Stokes solver with a finite-element structural solver. The optimization reduces the weighted drag coefficient by 1.2% over lift coefficients between 0.2 and 0.6, and increases the cavitation inception speed by 82% at the nominal condition compared to the baseline. The optimized design shows higher cavitation inception speeds than an Eppler hydrofoil (E1127, known for cavitation-free operation at moderate lift conditions) with the same baseline planform at CL≥0.2. The improvement from the E1127 hydrofoil was due mostly to delaying tip vortex cavitation induced by 3-D effects. The optimization finds the fiber angle to balance the bend-twist coupling and modifies the directional strength to reduce the susceptibility to excessive deformation and material failure when sweep presents.

Balancing Laminar Extension and Wave Drag for Transonic Swept Wings

At a transonic condition, the design of a natural laminar flow (NLF) wing is challenging because the extension of the laminar flow needs to be finely balanced with the potential wave drag increase. To achieve this balance, it is proposed to unlock the wing sweep and introduce a three-dimensional (3-D) contour shock-control bump (SCB) in the optimization of the NLF infinite swept wing aiming at total drag reduction. A 3-D Reynolds-averaged Navier–Stokes flow solver is extended to incorporate laminar–turbulent transition prediction due to streamwise and crossflow instabilities. The flow solver is integrated in a gradient-based optimization framework. The transition criteria including streamwise and crossflow instabilities are coupled in the sensitivity calculation using a discrete-adjoint solver. Transonic design optimization of the sectional profile and wing sweep angle is first conducted at a Mach number of 0.78. The optimization managed to alleviate the shock wave, reducing the pressure drag while it failed to extend the laminar flow. Furthermore, a combined optimization of the wing with a parameterized SCB at the same condition is carried out. The optimized wing with the SCB features a long favorable pressure gradient region and a low-sweep angle, which allows for a large proportion of laminar flow without significant pressure drag. The shock wave is controlled by the 3-D bump, and the SCB has little effect on the upstream flow.

Gradient-Based Aerodynamic Optimization of an Airfoil with Morphing Leading and Trailing Edges

This article presents a gradient-based aerodynamic optimization framework and investigates optimum deformations for a transonic airfoil equipped with morphing leading and trailing edges. Specifically, the proposed optimization framework integrates an innovative morphing shape parameterization with a high fidelity Reynolds-averaged Navier–Stokes computational fluid dynamic solver, a hybrid mesh deformation algorithm, and an efficient gradient evaluation method based on continuous adjoint implementation. To achieve a feasible morphing shape, some structural properties of skin and wing-box constraints were introduced into the morphing shape parameterization, which offers skin length control and enables wing-box shape invariance. In this study, the optimum leading and trailing edge deformations with minimization of drag at this cruise stage were searched for using the adjoint-based optimization with a nested feasible morphing procedure, subject to the wing-box, skin length, and airfoil volume constraints. The numerical studies verified the effectiveness of the optimization strategy, and demonstrated the significant aerodynamic performance improvement achieved by using the morphing devices. A lambda shock pattern was observed for the optimized morphing leading edge. That result further indicates the importance of leading edge radius control.

DAFoam: An Open-Source Adjoint Framework for Multidisciplinary Design Optimization with OpenFOAM

The adjoint method is an efficient approach for computing derivatives that allow gradient-based optimization to handle systems parameterized with a large number of design variables. Despite this advantage, implementing the adjoint method for a partial-differential-equation-based primal solver is a time-consuming task. To lower the barrier for adjoint implementations, an object-oriented framework (DAFoam) is proposed to rapidly implement the discrete adjoint method for any steady-state OpenFOAM primal solver by adding or modifying only a few hundred lines of source code. In this paper, the DAFoam framework is introduced and the proposed object-oriented adjoint development process is illustrated. Using this strategy, the adjoint method is implemented for eight primal solvers, five turbulence models, and one radiation model in OpenFOAM. Excellent adjoint speed and scalability, with up to 10 million cells and 1536 CPU cores, and an average error in the adjoint derivatives of less than 0.1% are achieved. Finally, the implemented adjoint solvers and models are integrated into a gradient-based optimization framework, and four distinct design optimizations (multipoint aerodynamic optimization of a low-speed unmanned-aerial-vehicle wing, aerodynamic optimization of a transonic aircraft configuration, aerothermal optimization of a turbine internal cooling passage, and aerostructural optimization of a compressor rotor) are shown. DAFoam is available under an open-source license and is a powerful tool for the high-fidelity multidisciplinary design optimization of engineering systems, such as aircraft, ground vehicles, marine vessels, and turbomachinery.