Results for “sensitivity gradient” — Page 15

NASA NTRS2020

SCoBi Multilayer: A Signals of Opportunity Reflectometry Model for Multilayer Dielectric Reflections

A multilayer module is incorporated into the Signals of Opportunity (SoOp) Coherent Bistatic Scattering model (SCoBi) for determining the reflections and propagation of electric fields within a series of multilayer dielectric slabs. This module can be used in conjunction with other SCoBi components to simulate complex, bistatic simulation schemes that include features such as surface roughness, vegetation, antenna effects, and multilayer soil moisture interactions on reflected signals. This paper introduces the physics underlying the multilayer module and utilizes it to perform a simulation study of the response of SoOp-R measurements with respect to subsurface soil moisture parameters. For a frequency range of 100–2400 MHz, it is seen that the SoOp-R response to a single dielectric slab is mostly frequency insensitive; however, the SoOp-R response to multilayer dielectric slabs will vary between frequencies. The relationship between SoOp-R reflectivity and the contributing depth is visualized, and the results show that SoOp-R measurements can display sensitivity to soil moisture below the penetration depth. By simulation of simple soil moisture profiles with different wetting and drying gradients, the dielectric contrast between layers is shown to be the greatest contributing factor to subsurface soil moisture sensitivity. Overall, it is observed that different frequencies can sense different areas of a soil moisture profile, and this behavior can enable subsurface soil moisture data products from SoOp-R observations.

Dylan Boyd↗

DOE OSTI2022

Automatic Tuner for the Step Sizes of Gradient-Based RT-OPF DERMS Control Systems [SWR-22-50]

Gradient-based controllers have parameters called step sizes that determine how sensitive its control actions are to received inputs. Finding a practical setting for a step size is important. If it is too small, the controller is ineffective by not reacting in a significant manor to its received inputs. On the other hand, if it is too large, the actions implemented by the controller may be excessive and cause the controlled system to become unstable. The main objective of the Automatic Tuner for the Step Sizes of Gradient-Based RT-OPF DERMS Control Systems is to speed up the implementation of gradient-based control schemes by automatically tuning its step sizes instead of having a practitioner spend considerable time doing the task manually. Only simple adjustments need to be made to the code of each controller with a step size to allow the addition of an individual automatic tuner. This can be applied to any RT-OPF based control for DER management.

Comden, Joshua↗

DOE OSTI2026

Hydrogeological assessment of CO2 containment assurance and wellbore integrity at a Gulf Coast storage site

Abstract A large-scale carbon capture and storage (CCS) initiative on the Texas Gulf Coast serves as a premier demonstration of the U.S. Department of Energy’s CarbonSAFE program. Targeting deep saline formations, specifically Oligo-Miocene deltaic sequences, the project aims to establish technical and commercial viability for geologic CO2 storage within a major industrial corridor. This study provides a rigorous hydrogeological assessment to support Class VI permitting by quantifying the high degree of containment security. Utilizing a compositional reservoir simulator, we developed a suite of 27 distinct simulation cases to evaluate vertical plume dynamics near both planned injection wells and proximal legacy infrastructure. To ensure numerical accuracy near wellbores, we implemented a refined mesh strategy, determining that a 5.6 ft × 5.6 ft grid refinement offered the optimal balance between computational efficiency and descriptive precision. The modeling framework utilized a systematic sensitivity-based approach to evaluate the mechanical redundancy of the subsurface system by performing a bounding analysis of wellbore interfaces against hypothetical high-permeability microannuli. By systematically isolating competing physical drivers, including permeability, porosity, gas hysteresis, thermal gradients, salinity, and solubility trapping (quantified via Henry’s law with dynamically adjusted coefficients), this work moves beyond binary assessments to establish a nuanced hierarchy of containment factors. The results confirm that primary trapping mechanisms (e.g., gas hysteresis and solubility), combined with the site's unique geomechanical stratigraphy, significantly restrict vertical mobility and reinforce the robust containment security of the reservoir. Baseline results demonstrate substantial vertical separation between the CO2 plume and the upper confining system, ensuring robust containment. Sensitivity analysis reveals that even under highly conservative bounding scenarios—assuming theoretical 10-Darcy pathways at specific wellbore locations—the 2,900-ft thick multi-layered confining zone remains a reliable barrier. In these hypothetical upper-bound cases, peak upward fluxes of CO2 and saltwater after 15 years of injection remain localized and dissipate rapidly within the lower sections of the confining interval, leaving the integrity of the seal uncompromised. Furthermore, the study identifies that while localized wellbore pathways define theoretical upper bounds of vertical migration, the Area of Review (AoR) is primarily sensitive to regional thermal gradients and hysteresis, which can influence the AoR by over 3,000 acres in pessimistic configurations. Also, primary trapping mechanisms, specifically gas hysteresis and solubility, work in tandem with the Gulf Coast’s unique geomechanical stratigraphy to significantly restrict vertical mobility. Ductile, smectite-rich mudstones facilitate natural borehole convergence and the self-healing of potential conduits, creating a natural geomechanical bridge that effectively mitigates migration potential at both current injection points and legacy-well locations. This comprehensive modeling effort demonstrates that the integration of high-resolution wellbore simulations and regional geomechanical observations confirms the long-term storage security of the studied site, providing a physics-based foundation for industrial-scale CCS deployments. This modeling framework establishes a baseline for future research into coupled geomechanical effects, such as time-dependent borehole convergence, to further refine long-term containment projections. Acknowledgements We thank the Gulf Coast Carbon Center (GCCC) at the Bureau of Economic Geology for foundational research support. We appreciate Alex Bump for technical guidance and David Hoffman for model mesh generation. This work used TACC’s Frontera cluster for simulations and CMG Ltd. software licenses provided to UT-Austin. This material is based upon work supported by the Department of Energy under Award Number DE-FE0032338. Disclaimer This material is based upon work supported by the U.S. Department of Energy’s Fossil Energy and Carbon Management Office under the CarbonSAFE program, award Number DE-FE0032338. The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

58 GEOSCIENCES↗

DOE OSTI2026

Hydrogeological assessment of CO2 containment assurance and wellbore integrity at a Gulf Coast storage site

Abstract A large-scale carbon capture and storage (CCS) initiative on the Texas Gulf Coast serves as a premier demonstration of the U.S. Department of Energy’s CarbonSAFE program. Targeting deep saline formations, specifically Oligo-Miocene deltaic sequences, the project aims to establish technical and commercial viability for geologic CO2 storage within a major industrial corridor. This study provides a rigorous hydrogeological assessment to support Class VI permitting by quantifying the high degree of containment security. Utilizing a compositional reservoir simulator, we developed a suite of 27 distinct simulation cases to evaluate vertical plume dynamics near both planned injection wells and proximal legacy infrastructure. To ensure numerical accuracy near wellbores, we implemented a refined mesh strategy, determining that a 5.6 ft × 5.6 ft grid refinement offered the optimal balance between computational efficiency and descriptive precision. The modeling framework utilized a systematic sensitivity-based approach to evaluate the mechanical redundancy of the subsurface system by performing a bounding analysis of wellbore interfaces against hypothetical high-permeability microannuli. By systematically isolating competing physical drivers, including permeability, porosity, gas hysteresis, thermal gradients, salinity, and solubility trapping (quantified via Henry’s law with dynamically adjusted coefficients), this work moves beyond binary assessments to establish a nuanced hierarchy of containment factors. The results confirm that primary trapping mechanisms (e.g., gas hysteresis and solubility), combined with the site's unique geomechanical stratigraphy, significantly restrict vertical mobility and reinforce the robust containment security of the reservoir. Baseline results demonstrate substantial vertical separation between the CO2 plume and the upper confining system, ensuring robust containment. Sensitivity analysis reveals that even under highly conservative bounding scenarios—assuming theoretical 10-Darcy pathways at specific wellbore locations—the 2,900-ft thick multi-layered confining zone remains a reliable barrier. In these hypothetical upper-bound cases, peak upward fluxes of CO2 and saltwater after 15 years of injection remain localized and dissipate rapidly within the lower sections of the confining interval, leaving the integrity of the seal uncompromised. Furthermore, the study identifies that while localized wellbore pathways define theoretical upper bounds of vertical migration, the Area of Review (AoR) is primarily sensitive to regional thermal gradients and hysteresis, which can influence the AoR by over 3,000 acres in pessimistic configurations. Also, primary trapping mechanisms, specifically gas hysteresis and solubility, work in tandem with the Gulf Coast’s unique geomechanical stratigraphy to significantly restrict vertical mobility. Ductile, smectite-rich mudstones facilitate natural borehole convergence and the self-healing of potential conduits, creating a natural geomechanical bridge that effectively mitigates migration potential at both current injection points and legacy-well locations. This comprehensive modeling effort demonstrates that the integration of high-resolution wellbore simulations and regional geomechanical observations confirms the long-term storage security of the studied site, providing a physics-based foundation for industrial-scale CCS deployments. This modeling framework establishes a baseline for future research into coupled geomechanical effects, such as time-dependent borehole convergence, to further refine long-term containment projections. Acknowledgements We thank the Gulf Coast Carbon Center (GCCC) at the Bureau of Economic Geology for foundational research support. We appreciate Alex Bump for technical guidance and David Hoffman for model mesh generation. This work used TACC’s Frontera cluster for simulations and CMG Ltd. software licenses provided to UT-Austin. This material is based upon work supported by the Department of Energy under Award Number DE-FE0032338. Disclaimer This material is based upon work supported by the U.S. Department of Energy’s Fossil Energy and Carbon Management Office under the CarbonSAFE program, award Number DE-FE0032338. The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

58 GEOSCIENCES↗

DOE OSTI2026

Response of marine post-frontal clouds to Gulf Stream variability

Understanding how Gulf Stream variation influences cloud morphology is critical for evaluating cloud feedback in the western North Atlantic Ocean and beyond, where mesoscale air-sea interactions dominate. This study investigates the impact of altered mean sea surface temperature (SST) and SST gradients on post-frontal cloud characteristics during cold-air outbreaks, using the Weather Research and Forecasting (WRF) model. Three sensitivity experiments are conducted: a control simulation (default SST), Plus4 (uniform SST increase of 4 K), and Gradplus (SST gradient enhanced by 25 %, centered around mean SST). Results reveal distinctly different responses in boundary layer dynamics and cloud macro-physics. In Plus4, a warmer and moister boundary layer reduces total cloud cover but promotes larger cloud sizes and elongated cloud streets, with diminished liquid water and enhanced ice-phase hydrometeors. Conversely, Gradplus amplifies impacts in the upwind colder SST regions, yielding a drier, colder boundary layer, weaker energy transport, and higher liquid water path but reduced ice water content and cloud lines. Tracer analysis highlights that SST modifications alter airmass sources near cloud tops due to the entrainment of ambient air, with Plus4 amplifying boundary layer contributions to cloud-top regions. These findings underscore the spatially varying effects of SST gradients and mean SST on cloud organization and microphysics, emphasizing the need to resolve ocean-atmosphere coupling in global models to improve the prediction of marine cloud feedback under warming scenarios.

AIR-SEA INTERACTION↗

DOE OSTI2021

Multiscale design of nonlinear materials using a Eulerian shape optimization scheme

Motivated by recent advances in manufacturing, the design of materials is the focal point of interest in the material research community. One of the critical challenges in this field is finding optimal material microstructure for a desired macroscopic response. This work presents a computational method for the mesoscale-level design of particulate composites for an optimal macroscale-level response. The method relies on a custom shape optimization scheme to find the extrema of a nonlinear cost function subject to a set of constraints. Three key “modules” constitute the method: multiscale modeling, sensitivity analysis, and optimization. Multiscale modeling relies on a classical homogenization method and a nonlinear NURBS-based generalized finite element scheme to efficiently and accurately compute the structural response of particulate composites using a nonconformal discretization. A three-parameter isotropic damage law is used to model microstructure-level failure. An analytical sensitivity method is developed to compute the derivatives of the cost/constraint functions with respect to the design variables that control the microstructure's geometry. The derivation uncovers subtle but essential new terms contributing to the sensitivity of finite element shape functions and their spatial derivatives. Several structural problems are solved to demonstrate the applicability, performance, and accuracy of the method for the design of particulate composites with a desired macroscopic nonlinear stress-strain response.

42 ENGINEERING↗

DOE OSTI2025

Response tailoring of elasto-plastic trusses

Abstract In this work we tailor the response of trusses loaded beyond their yield limit. The truss structures are modeled using finite strain theory and rate-independent elasto-plasticity. We design trusses with a tailored mechanical response that is between “elastic” and “elastic-ideal-plastic” subject to the volume constraint. The design updates are generated by the gradient-based Method of Moving Asymptotes (MMA) solver and the sensitivities of the response functions are computed using a path-dependent adjoint sensitivity analysis. The computations are performed in Matlab.

42 ENGINEERING↗

DOE OSTI2021

Elemental segregation to lattice defects in the CrMnFeCoNi high-entropy alloy during high temperature exposures

The influence of small plastic pre-strains on the elevated-temperature stability and microstructure of the equiatomic CrMnFeCoNi FCC solid solution is investigated. Particular attention is given to whether any of the alloy elements segregate to individual dislocations. To that end, CrMnFeCoNi samples were first deformed in tension at room temperature to plastic strains of 0.2 and 2.3%, and subsequently annealed at 973 K for 800 hours. The pre-strains activated planar slip of 1/2<110>-type dislocations on {111}-type glide planes. Interactions of this planar slip with special Σ3 grain boundaries formed a large number of dislocation segments with a <110>-type crystallographic orientation suitable for a credible end-on analysis of dislocation cores in HR-STEM. The cores of the 1/2<110> dislocations pushed up against the investigated grain boundaries were found to be close to the compact configuration. Within the sensitivity of the Super-X EDS mapping, no concentration gradient was detected near dislocations that would indicate enrichment at dislocation cores of any of the elemental constituents of the alloy after the pre-deformation and annealing. However, a Cr-rich tetragonal sigma phase nucleated and grew at grain boundary triple junctions during this anneal, processes that were not accelerated by the enhanced dislocation density present after pre-strain. A clear chromium gradient was observed in the Cr-depleted zones near grain boundaries suggesting that Cr transport occurred by relatively slow diffusion from the bulk to the grain boundaries and then by relatively fast diffusion along the grain boundaries to the precipitates. Accompanying the Cr depletion near grain boundaries is a simultaneous Ni and Mn enrichment, which promotes formation of the L1 0 NiMn phase that is observed on the grain boundaries after prolonged annealing.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Co-Active Subspace Methods for the Joint Analysis of Adjacent Computer Models

Active subspace (AS) methods are a valuable tool for understanding the relationship between the inputs and outputs of a Physics simulation. In this article, an elegant generalization of the traditional ASM is developed to assess the co-activity of two computer models. This generalization, which we refer to as a Co-Active Subspace (Co-AS) Method, allows for the joint analysis of two or more computer models allowing for thorough exploration of the alignment (or non-alignment) of the respective gradient spaces. We define co-active directions, co-sensitivity indices, and a scalar “concordance” metric (and complementary “discordance” pseudo-metric) and we demonstrate that these are powerful tools for understanding the behavior of a class of computer models, especially when used to supplement traditional AS analysis. Details for efficient estimation of the Co-AS and an accompanying R package (concordance) are provided. Practical application is demonstrated through analyzing a set of simulated rate stick experiments for PBX 9501, a high explosive, offering insights into complex model dynamics.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Anomalous thermal effect in Zr Te 5 observed via photothermal measurements

In this study, we explore the magneto-thermoelectric power (MTP) of $ZrTe$ 5 , a canonical Dirac semimetal, through a novel photothermal technique. Unlike conventional thermoelectric studies that rely on on-chip heaters and are limited by fabrication processes, especially for stress-sensitive materials, our approach utilizes photothermal effects to induce temperature gradients. Our experiments, applying a magnetic field approximately parallel and transverse to the photocurrent detection direction, reveal that the photothermal method efficiently and reliably extracts both diagonal and off-diagonal components of the thermoelectric coefficient of $ZrTe$ 5 . Here, we observe that the longitudinal MTP reproduces features previously reported in thermal transport studies, while the photoinduced transverse MTP confirms the anomalous Nernst effect. This photothermal measurement technique opens new avenues for investigating transport properties in a wide range of quantum materials, both in 3D and 2D systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Multi-Federate Co-Convergence with HELICS

In co-simulation studies, convergence refers to the ability of different simulation tools involved to achieve a consistent and stable solution. Convergence is a critical aspect of co-simulation as it determines the accuracy of the results obtained from the simulation. Convergence in co-simulation studies depends on several factors, this includes system complexity, the accuracy of the models used, and the numerical methods employed by the simulation tools. It is essential to ensure that the coupling interfaces between the different tools are designed to allow for data exchange in a consistent and accurate manner. Hierarchical Engine for Large-scale Infrastructure Co-Simulation (HELICS) is an open-source co-simulation framework developed for the energy domain. This paper explores the convergence performance of a set of co-simulation use cases. We further explore the use of a co-convergence helper federate to help with co-simulation convergence. The convergence efficacy of several algorithms (both gradient-based and gradient-free) is tested against these use cases. Finally, the sensitivity of these algorithms to several factors, such as system scaling and others, is tested and detailed in this paper. Our results show that for a subset of use cases, the co-convergence helper federate is able to improve co-simulation convergence significantly.

co-convergence↗

DOE OSTI2025

Tropospheric aerosols over the western North Atlantic Ocean during the winter and summer deployments of ACTIVATE 2020: life cycle, transport, and distribution

The Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) is a NASA mission to characterize aerosol–cloud interactions over the western North Atlantic Ocean (WNAO). Such characterization requires understanding of life cycle, composition, transport pathways, and distribution of aerosols over the WNAO. This study uses the GEOS-Chem model to simulate aerosol distributions and properties that are evaluated against aircraft, ground-based, and satellite observations during the winter and summer field deployments in 2020 of ACTIVATE. Transport in the boundary layer (BL) behind cold fronts was a major mechanism for the North American continental outflow of pollution to the WNAO in winter. Turbulent mixing was the main driver for the upward transport of sea salt within and ventilation out of BL in winter. The BL aerosol composition was dominated by sea salt, which increased in the summer, followed by organics and sulfate. Aircraft in situ aerosol measurements provided useful constraints on wet scavenging in GEOS-Chem. The model generally captured observed features such as continental outflow, land–ocean gradient, and mixing of anthropogenic aerosols with sea salt. Model sensitivity experiments with elevated smoke injection heights to the mid-troposphere (versus within BL) better reproduced observations of smoke aerosols from the western US wildfires over the WNAO in the summer. Model analysis suggests strong hygroscopic growth of sea salt particles and their seeding of marine BL clouds over the WNAO (< 35° N). Future modeling efforts should focus on improving parameterizations for aerosol wet scavenging, implementing realistic smoke injection heights, and applying high-resolution models that better resolve vertical transport.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1976

Interplanetary stream magnetism - Kinematic effects

The particle density and the magnetic-field intensity and direction are calculated for volume elements of the solar wind as a function of the initial magnetic-field direction and the initial speed gradient. It is assumed that the velocity is constant and radial. These assumptions are approximately valid between about 0.1 and 1.0 AU for many streams. Time profiles of the particle density, field intensity, and velocity are calculated for corotating streams, neglecting effects of pressure gradients. The compression and rarefaction of the magnetic field depend sensitively on the initial field direction. By averaging over a typical stream, it is found that the average radial field intensity is inversely proportional to the square of the heliocentric distance, whereas the average intensity in the direction of the planets' motion does not vary in a simple way, consistent with deep space observations. Changes of field direction may be very large, depending on the initial angle; but when the initial angle at 0.1 AU is such that the base of the field line corotates with the sun, the spiral angle is the preferred direction at 1 AU. The theory is also applicable to nonstationary flows.

Burlaga, L. F.↗

NASA NTRS1985

Low Reynolds number separation bubble research at UTRC

The facilities and techniques being applied in investigations of laminar separation bubbles on low Re airfoils at the United Technologies Research Center are described. The research is focused on developing a database and predictive models for use in designing compressor, fan and turbine blades. Flow data are gathered primarily by laser Doppler velocimetry. Experimental procedures are being devised which will permit formation of a short bubble for characterizations of the velocity profiles in and around the bubble and boundary layer. Further tests are directed at defining the size and location of the bubble and the transition in the separated shear layer. The sensitivity of the bubble to alterations in the Re, pressure gradient, turbulence intensity and the turbulence scales will be examined. The effects of surface curvature and roughness on the bubble will also be studied. Details of the wind tunnel facility and the steps being taken to define conditions for reliably producing the short bubble are outlined.

Patrick, W. P.↗

NASA NTRS1999

A Novel Shape Parameterization Approach

This paper presents a novel parameterization approach for complex shapes suitable for a multidisciplinary design optimization application. The approach consists of two basic concepts: (1) parameterizing the shape perturbations rather than the geometry itself and (2) performing the shape deformation by means of the soft objects animation algorithms used in computer graphics. Because the formulation presented in this paper is independent of grid topology, we can treat computational fluid dynamics and finite element grids in a similar manner. The proposed approach is simple, compact, and efficient. Also, the analytical sensitivity derivatives are easily computed for use in a gradient-based optimization. This algorithm is suitable for low-fidelity (e.g., linear aerodynamics and equivalent laminated plate structures) and high-fidelity analysis tools (e.g., nonlinear computational fluid dynamics and detailed finite element modeling). This paper contains the implementation details of parameterizing for planform, twist, dihedral, thickness, and camber. The results are presented for a multidisciplinary design optimization application consisting of nonlinear computational fluid dynamics, detailed computational structural mechanics, performance, and a simple propulsion module.

Samareh, Jamshid A.↗

NASA NTRS2000

Multidisciplinary Aerodynamic-Structural Shape Optimization Using Deformation (MASSOUD)

This paper presents a multidisciplinary shape parameterization approach. The approach consists of two basic concepts: (1) parameterizing the shape perturbations rather than the geometry itself and (2) performing the shape deformation by means of the soft object animation algorithms used in computer graphics. Because the formulation presented in this paper is independent of grid topology, we can treat computational fluid dynamics and finite element grids in a similar manner. The proposed approach is simple, compact, and efficient. Also, the analytical sensitivity derivatives are easily computed for use in a gradient-based optimization. This algorithm is suitable for low-fidelity (e.g., linear aerodynamics and equivalent laminated plate structures) and high-fidelity (e.g., nonlinear computational fluid dynamics and detailed finite element modeling analysis tools. This paper contains the implementation details of parameterizing for planform, twist, dihedral, thickness, camber, and free-form surface. Results are presented for a multidisciplinary design optimization application consisting of nonlinear computational fluid dynamics, detailed computational structural mechanics, and a simple performance module.

Samareh, Jamshid A.↗

NASA NTRS2000

Multidisciplinary Aerodynamic-Structural Shape Optimization Using Deformation (MASSOUD)

This paper presents a multidisciplinary shape parameterization approach. The approach consists of two basic concepts: (1) parameterizing the shape perturbations rather than the geometry itself and (2) performing the shape deformation by means of the soft object animation algorithms used in computer graphics. Because the formulation presented in this paper is independent of grid topology, we can treat computational fluid dynamics and finite element grids in the same manner. The proposed approach is simple, compact, and efficient. Also, the analytical sensitivity derivatives are easily computed for use in a gradient-based optimization. This algorithm is suitable for low-fidelity (e.g., linear aerodynamics and equivalent laminate plate structures) and high-fidelity (e.g., nonlinear computational fluid dynamics and detailed finite element modeling) analysis tools. This paper contains the implementation details of parameterizing for planform, twist, dihedral, thickness, camber, and free-form surface. Results are presented for a multidisciplinary application consisting of nonlinear computational fluid dynamics, detailed computational structural mechanics, and a simple performance module.

Samareh, Jamshid A.↗

NASA NTRS1997

Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

Anderson, W. Kyle↗