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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Analysis of the SiMPL Method for Density-Based Topology Optimization

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides pointwise bound-preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [D. Kim et al., Struct. Multidiscip. Optim., 68 (2025), 74]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. Here, we refer to the new algorithm as the SiMPL method (pronounced “simple”), which stands for Sigmoidal Mirror descent with a Projected Latent variable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A simple introduction to the SiMPL method for density-based topology optimization

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as “the simple method”) optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. In conclusion, to outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Calculus of Variations and Optimization↗
DOE OSTI2025

Multi-physics Topology OPtimization and Additive Manufacturing for High-temperature Heat Exchangers

This research significantly advances the understanding of high-temperature heat exchanger design through an integrated approach that combines topology optimization (TO), triply periodic minimal surface (TPMS) structures, additive manufacturing (AM) and thermohydraulic testing. Each of these components contributes uniquely to a unified, high-performance design, fabrication and testing workflow. Topology optimization serves as the foundation of the design methodology by providing a systematic way to determine the most effective material layout for separating hot and cold fluids while maximizing thermal performance. The researchers introduced a novel three-material optimization framework using two density fields to represent hot fluid, cold fluid, and solid domains. This approach enables automated discovery of optimal shapes and flow paths that cannot be intuitively designed, especially under constraints imposed by manufacturing technologies. Furthermore, constraints such as minimal wall thickness and overhang angles were embedded into the optimization process, ensuring that resulting designs are not only thermally efficient but also manufacturable using modern additive techniques. In parallel, the study delves into the use of Gyroid-based TPMS geometries for constructing the core of the heat exchanger. TPMS structures are known for their high surface area, excellent fluid mixing capabilities, and minimal pressure drop characteristics. The researchers applied a data-driven modeling framework using Heteroscedastic Sparse Gaussian Process Regression (HSGPR) combined with genetic algorithms. This allowed for the rapid evaluation and optimization of key geometric parameters such as frequency, iso-value, and phase shift. The result was a set of Gyroid structures tailored for high heat transfer and low flow resistance, demonstrating clear improvements over conventional straight-channel designs. After the designing process, additive manufacturing played a critical role by turning these highly complex, optimized geometries into physical components. Utilizing Laser Powder Bed Fusion (LPBF) with Haynes 282, the study demonstrated the feasibility of fabricating these heat exchangers at high precision. Post-processing methods, including dilation-erosion operations, were applied to ensure local features adhered to self-supporting constraints. The fabricated structures were then subjected to thermohydraulic testing under conditions representative of supercritical CO 2 Brayton cycles, validating the predicted performance and confirming the viability of the full design-to-fabrication pipeline. Finally, thermohydraulic testing across the above studies served as a crucial experimental validation of advanced heat exchanger. Under consistent high-temperature and high-pressure conditions using supercritical CO 2 , the testing demonstrated that both TO and Gyroid-based TPMS designs significantly outperformed conventional straight-channel HXs. The TO design achieved a 115% increase in UA and NTU and a 27.6% boost in gravimetric power density, while the data-driven optimized Gyroid design delivered a 166% increase in UA and NTU and improved effectiveness from 68.7% to 86.1%. These results validate the simulation models, confirm the manufacturability of complex geometries under AM constraints, and provide key insights into design-performance trade-offs, thereby advancing the development of high-efficiency, compact heat exchangers for extreme environments.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Level-set topology optimization with PDE generated conformal meshes

This paper presents a level-set topology optimization approach that uses conformal meshes for the analysis of the displacement field. The structure’s boundary is represented by the iso-contour of a level-set field discretized on a fixed background design mesh. The conformal mesh is updated for each design iteration via a PDE based mesh morphing process that identifies the set of facets in the background mesh that are homeomorphic to the boundary and relaxes the homeomorphic mesh to conform to the structure’s boundary and ensure high element quality. The conformal mesh allows for a more accurate computation of the response versus density and some level-set based methods which interpolate material properties using the volume fraction. Numerical examples illustrate the proposed approach by optimizing linear-elastic two- and three-dimensional structures, wherein insight into the performance of the mesh morphing process is provided. The examples also highlight the scalability of the approach.

42 ENGINEERING↗
DOE OSTI2025

Advanced Transmission Technologies – GETs and HPCs Session 2: Advanced Power Flow Control and Transmission Topology Optimization

The INL TADA GETs Cohort Session 2, held on November 7, 2025, conducted in collaboration with ScottMadden, focused on two core Advanced Transmission Technologies (ATTs): Advanced Power Flow Control (APFC) and Transmission Topology Optimization (TTO). These technologies are pivotal in enhancing grid flexibility, reliability, and cybersecurity resilience. APFC, particularly through modular FACTS devices like Modular Static Synchronous Series Compensators (M-SSSCs), enables dynamic voltage injection to reroute power flows. The session highlighted the deployment benefits of APFC, such as rapid installation, minimal civil works, and re-deployability. Regulatory drivers like FERC Order 2023 mandate the inclusion of Grid-Enhancing Technologies (GETs) in interconnection studies. Case studies from Central Hudson, CAISO, and National Grid (UK) demonstrated APFC’s effectiveness in congestion relief and cost savings. The session also addressed cybersecurity concerns, including firmware vulnerabilities, SCADA integration risks, and supply chain dependencies. Participants engaged in interactive exercises to rank cybersecurity and supply chain risks, emphasizing the need for robust digital assurance strategies. TTO involves software-based reconfiguration of transmission networks to optimize power flow without new infrastructure. The session showcased its operational value, with examples from SPP, PJM, and MISO showing significant congestion cost reductions. Cybersecurity vulnerabilities were discussed, particularly in API security and software supply chains, referencing incidents like SolarWinds and attacks on Danish utilities. Digital assurance exercises explored worst-case scenarios, attack paths, and mitigation responsibilities between vendors and utilities. Reliability challenges such as algorithm stability, vendor dependency, and operator trust were also examined. Cross-cutting themes emphasized the importance of digital assurance tools, including Software Bills of Materials (SBOMs) and hardware-in-loop testing. Human performance, training, and operational confidence were identified as critical enablers of technology adoption. The session concluded with a preview of Session 3, which will focus on High Performance Conductors (HPCs) and risk-based cybersecurity tools. Session 2 of 3.

24 - POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Adaptive immersed isogeometric level-set topology optimization

Here, this paper presents for the first time an adaptive immersed approach for level-set topology optimization using higher-order truncated hierarchical B-spline discretizations for design and state variable fields. Boundaries and interfaces are represented implicitly by the iso-contour of one or multiple level-set functions. An immersed finite element method, the eXtended IsoGeometric Analysis, is used to predict the physical response. The proposed optimization framework affords different adaptively refined higher-order B-spline discretizations for individual design and state variable fields. The increased continuity of higher-order B-spline discretizations together with local refinement enables direct control over the accuracy of the representation of each field while simultaneously reducing computational cost compared to uniformly refined discretizations. A flexible mesh adaptation strategy enables local refinement based on geometric measures or physics-based error indicators. These adaptive discretization and analysis approaches are integrated into gradient-based optimization schemes, evaluating the design sensitivities using the adjoint method. Numerical studies illustrate the features of the proposed framework with static, linear elastic, multi-material, two- and three-dimensional problems. The examples provide insight into the effect of refining the design variable field on the optimization result and the convergence rate of the optimization process. Using coarse higher-order B-spline discretizations for level-set fields promotes the development of smooth designs and suppresses the emergence of small features. Moreover, adaptive mesh refinement for state variable fields results in a reduction of overall computational cost. Higher-order B-spline discretizations are especially interesting when evaluating gradients of state variable fields due to their higher inter-element continuity.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Topology Optimization of Curved Electrodes with Suggested Simplified Corrugated Designs

The demand for space-efficient energy storage, from grid to device scale, has soared with the rapid electrification of society. To increase energy density while maintaining high rate performance, shaping electrodes has been shown to be an effective strategy. Computational optimization can suggest designs that improve energy storage performance; however, generated designs are often complex, leading to manufacturing challenges. Additionally, most optimization is typically conducted in standard rectangular domains, whereas some applications, such as in the aerospace industry or in consumer electronics, may benefit from optimized electrodes that fit in a nonstandard form factor. Here, in this work, we use topology optimization to design capacitive full-cell electrodes in a curved domain. We propose a simplified corrugated electrode design based on the optimization results and show that the energy stored in the simplified design is only reduced by about 5–27% when compared to that in the fully optimized design, with more deviation occurring when ionic transport in the porous electrode is particularly impeded. Overall, we find that these simplified designs are promising, particularly given the more conventional manufacturing techniques required, and have a 42–369% increase in energy stored compared to monolithic electrodes.

Energy - Storage↗
DOE OSTI2024

Topology optimization for the full-cell design of porous electrodes in electrochemical energy storage devices

In this paper, we introduce a density-based topology optimization framework to design porous electrodes for maximum energy storage. We simulate the full cell with a model that incorporates electronic potential, ionic potential, and electrolyte concentration. The system consists of three materials, namely pure liquid electrolyte and the porous solids of the anode and cathode, for which we determine the optimal placement. We use separate electronic potentials to model each electrode, which allows interdigitated designs. As a result, a penalization is required to ensure that the anode and cathode do not touch, i.e., causing a short circuit. We compare multiple 2D designs generated for different fixed conditions, e.g. material properties. A 3D design with complex channel and interlocked structure is also created. All optimized designs are far superior to the traditional monolithic electrode design with respect to energy storage metrics. We observe up to a 750% increase in energy storage for cases with slow effective ionic diffusion within the porous electrode.

25 ENERGY STORAGE↗
DOE OSTI2024

Enhancing the cooling performance of thermocouples: a power-constrained topology optimization procedure

Abstract Heat pumping through thermoelectric devices has many advantages over traditional cooling. However, their current efficiency is a limiting factor in their implementation. In this paper, we approach the non-convex topology optimization of thermoelectrical elements for cooling applications through the method of moving asymptotes (MMA) to improve their cooling capabilities per watt usage. The optimization problem is defined for a given power budget, aiming for the minimum temperature with a known heat pumping need. The introduction of power as a constraint justifies the introduction of the voltage gradient across the thermocouple as a design variable to maintain the thermoelectrical device in its optimum power-to-heat extraction ratio. To better understand the convergence of this non-convex problem, we present a two-variable analytical thermoelectric optimization model. This example provides information on how to select the penalty parameters used to scale the three material coefficients involved in the problem to obtain lower objective values and better convergence using MMA. The analytical model shows the non-convexity of the problem and provides the recommendation to use penalization coefficients of the form $$p_k=p_{\sigma }>p_{\alpha }=1$$ p k = p σ > p α = 1 for the thermal conductivity, electrical conductivity, and Seebeck coefficients. We tested these penalization coefficients through optimizations of a model based on the 1MC10-031 commercial thermoelectric-cooler (TEC) using the finite element method (FEM). These penalization coefficients provided local minima without the need for volume constraints. With this procedure, we found designs that provided temperatures close to 10 degrees lower using 60% less semiconductor material volume compared to the initial design.

Gutiérrez, G. Reales↗
DOE OSTI2024

Multi-Physics Topology Optimization for Dual Flow Heat Exchangers

Current heat exchanger (HX) designs are subjected to strict constraints pertaining to their manufacturability through conventional lengthy and expensive processes consisting of subtractive, forming and brazing processes. Therefore, existing HXs are typically restricted to the classical and rather simple geometries, such as plate-fin or shell-tube geometries, with lower efficiency and larger size. To increase HX efficiencies, novel designs are needed. Recent advances in topology optimization (TO) and additive manufacturing (AM) offer significant flexibility in design and fabrication of complex geometries, with potentially superior thermo-mechanical performance and high-power density. On the other hand, there is no commercially available software tool that addresses the combined coupling of structural, fluid dynamics, and thermal with multiple materials (hot and cold streams, solid, and void space) in TO of HXs. The goal of this project is to develop a multi-physics multi-phase TO approach to enhance the performance of HXs. The optimized design will be additively manufactured and tested in a physical environment to quantify its performance.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2026

Enabling On-Demand Aerospace Component Manufacturing: Topology Optimization of GE Engine Bracket and Fabrication Using Metal FFF

Additive Manufacturing (AM) offers advantages over conventional manufacturing processes, particularly by reducing the number of parts produced through multistage combined technologies, but these often result in low manufacturing yields or require post-processing. AM facilitates the production of complex geometries with fine features, overhangs, and lattice structures. For instance, Laser Powder Bed Fusion (LPBF) technology enables the fabrication of intricate parts that can be easily post-processed by removing residual powder. Laser powder bed AM technologies are widely discussed in the literature due to their design freedom in creating complex geometries, with and without the need for support generation. However, rapid solidification due to a thermal gradient in the build direction, which leads to the formation of columnar grains and warpage, is one of the challenges. To address this challenge, we propose layer-by-layer metal FFF technology, followed by the debinding and sintering process, as an alternative to powder- and laser-based approaches. Furthermore, design for additive manufacturing (DfAM) principles are discussed to minimize the need for support generation and enable easy post-processing, thereby improving surface finish to meet high tolerances in fabricating components for aerospace and healthcare applications.

Singh, Abhishek [University of Michigan, Ann Arbor↗
DOE OSTI2024

Optimize Topology, Component Sizes, and Operating Strategy of Participant's Protype: Cooperative Research and Development Final Report

The purpose of the project was to complete testing of Shine Technologies LLC’s prototype power system called “Juicebox 3.0.” Testing of the prototype using NREL’s hardware-in-the-loop capabilities, including a DC power supply programmed to simulate PV module output and provide the JuiceBox 3.0 at the specified power level and with a load bank of sufficient capacity to mimic DC and AC loads connected to the outlets of the Juicebox 3.0. The unit is provided with SAE wire connectors. A photo of a JuiceBox 3.0 unit is shown in Figure 1 below.

14 SOLAR ENERGY↗