Results for “Object Model” — Page 40

NASA NTRS1992

Noise-driven radiative shocks. II - Further implications for the quasi-periodic oscillations of the AM Herculis objects

The driven-shock model by Wolff et al. (1992) is extended and used to model the low-amplitude quasiperiodic oscillations (QPOs) driven by accretion noise in AM Her objects. The driven-shock model is discussed qualitatively with calculational details of the revised version. QPO-like features in the optical and X-ray are shown to be produced by bremsstrahlung-dominated shocks from accretion onto white dwarfs. These features are found to be consistent with the range of properties associated with the AM Her QPO sources. The bremsstrahlung-dominated shocks are shown to have linear responses for the conditions associated with QPO sources, and a luminosity expression for QPO sources is derived. The present analysis supports the shock-oscillation model for describing the optical QPOs from these objects.

Wood, Kent S.↗

DOE OSTI2022

APPLICATION OF OBJECTIVES-DRIVEN ASSURANCE CASES TO SYSTEM DEVELOPMENT IN AN EVOLVING ACQUISITION MODEL

System properties such as “safety” and “dependability” cannot, in practice, be proven, and must be argued in an “assurance case” aimed at supporting risk-acceptance decisions that have to be made by system acquirers and/or regulatory authorities. The paper is concerned with applications of the “assurance case” idea early in design and development of new systems, when (apart from dedicated testing) the only available operating experience information derives from previous (non-identical) systems. Much of the discussion is based on an evolving acquisition model at the US National Aeronautics and Space Administration; previously, most major systems were developed in-house, but some major systems will now be developed by and acquired from commercial providers. Key points discussed include the following. (1) By promoting a particular kind of focused discussion between acquirers and providers, the use of assurance cases should be particularly valuable under the new acquisition model. (2) In principle, objectives-driven (sometimes called “performance-based”) approaches to assurance of performance have significant advantages in cases where they are applicable. (3) For truly novel systems, completeness of the safety analysis is a significant issue; it is important for the assurance case to include a commitment by the provider (or applicant) to seriously pursue analysis of operating experience, so that previously unrecognized hazards can be identified and addressed. (4) Inquiries into major accidents often point to deficiencies in management oversight in all parts of the life cycle; management processes need to be addressed in the formulation and the implementation of an assurance case. Under the new acquisition model, these considerations imply a serious reconsideration of the way in which the development process is managed by both providers and acquirers.

99 GENERAL AND MISCELLANEOUS↗

NASA NTRS1987

A technique for 3-D robot vision for space applications

An extension of the MIAG algorithm for recognition and motion parameter determination of general 3-D polyhedral objects based on model matching techniques and using Moment Invariants as features of object representation is discussed. Results of tests conducted on the algorithm under conditions simulating space conditions are presented.

Markandey, V.↗

NASA NTRS1987

Demonstration of a 3D vision algorithm for space applications

This paper reports an extension of the MIAG algorithm for recognition and motion parameter determination of general 3-D polyhedral objects based on model matching techniques and using movement invariants as features of object representation. Results of tests conducted on the algorithm under conditions simulating space conditions are presented.

Defigueiredo, Rui J. P.↗

NASA NTRS2001

Modelling of Trapped Radiation Near Jupiter

Energetic (62 to approx. 130 MeV) proton fluxes measured with the Galileo Probe inside Jupiter's main ring (radii 122,500 to 128,940 km) show a modest increase as Jupiter is approached. Solutions of a reduced (1-d) equatorial diffusion equation with-constant losses described by a lifetime tau match those data for tau greater than or equal to 10(exp 9) s for diffusion coefficient D(sub LL) = 10(exp -9) L(exp 4)/s, 10(exp -9) L(exp -3)/s, or 10(exp -10) L(exp 4)/s (this particle population may be undergoing nearly loss-free inward radial diffusion). Tau would increase as Jupiter is approached if its value were determined by pitch angle scattering due to EM wave-particle interactions. if the absolute amplitudes of the waves' magnetic fluctuations did not vary with radial distance. Exploration of numerical solutions of the diffusive transport equation for Jupiter's magnetospheric ions at die planetary ring and closer to Jupiter has been done. Explicit range-energy relationships for energy loss in ring matter (SiO2), modeled as a continuous disk, are incorporated. A spatial resolution of approx. 1000 km in the radial direction is used in the vicinity of the main ring; off-equatorial ion fluxes have been included (assuming the atmosphere is a perfect absorber). For protons the maximum effect of energy loss in the microscopic size range of ring particles is expected at energies approx. 0.1 Mev. Presumably there could be sufficiently large, unobserved bodies within the ring that would have to be modeled as discrete objects, as planetary satellites would be. Within limitations of the model and computational resources it's planned to characterize solutions for transport of magnetospheric ions past the planetary ring; the procedures that are used are fairly well known and may be applied generally to the simpler magnetospheric models.

Mihalov, John D.↗

NASA NTRS2010

Modeling of Alkane Oxidation Using Constituents and Species

It is currently not possible to perform simulations of turbulent reactive flows due in particular to complex chemistry, which may contain thousands of reactions and hundreds of species. This complex chemistry results in additional differential equations, making the numerical solution of the equation set computationally prohibitive. Reducing the chemical kinetics mathematical description is one of several important goals in turbulent reactive flow modeling. A chemical kinetics reduction model is proposed for alkane oxidation in air that is based on a parallel methodology to that used in turbulence modeling in the context of the Large Eddy Simulation. The objective of kinetic modeling is to predict the heat release and temperature evolution. This kinetic mechanism is valid over a pressure range from atmospheric to 60 bar, temperatures from 600 K to 2,500 K, and equivalence ratios from 0.125 to 8. This range encompasses diesel, HCCI, and gas-turbine engines, including cold ignition. A computationally efficient kinetic reduction has been proposed for alkanes that has been illustrated for n-heptane using the LLNL heptane mechanism. This model is consistent with turbulence modeling in that scales were first categorized into either those modeled or those computed as progress variables. Species were identified as being either light or heavy. The heavy species were decomposed into defined 13 constituents, and their total molar density was shown to evolve in a quasi-steady manner. The light species behave either in a quasi-steady or unsteady manner. The modeled scales are the total constituent molar density, Nc, and the molar density of the quasi-steady light species. The progress variables are the total constituent molar density rate evolution and the molar densities of the unsteady light species. The unsteady equations for the light species contain contributions of the type gain/loss rates from the heavy species that are modeled consistent with the developed mathematical forms for the total constituent molar density rate evolution; indeed, examination of these gain/loss rates shows that they also have a good quasi-steady behavior with a functional form resembling that of the constituent rate. This finding highlights the fact that the fitting technique provides a methodology that can be repeatedly used to obtain an accurate representation of full or skeletal kinetic models. Assuming success with the modified reduced model, the advantage of the modeling approach is clear. Because this model is based on the Nc rate rather than on that of individual heavy species, even if the number of species increases with increased carbon number in the alkane group, providing that the quasi-steady rate aspect persists, then extension of this model to higher alkanes should be conceptually straightforward, although it remains to be seen if the functional fits would remain valid or would require reconstruction.

Bellan, Jasette↗

DOE OSTI2025

Optimization of a Secondary Air Injector for a Rich-Quench-Lean (RQL) Ammonia Combustor Using Computational Fluid Dynamics

Ammonia has emerged as a promising medium for moving hydrogen around the globe due to its energy density, existing infrastructure for production, transportation, and storage, and multiple applications from fertilizer to power generation. However, one of the most significant challenges with ammonia combustion for power generation is the formation of nitrogen oxides (NOx) during combustion. Several combustion strategies have been developed to minimize the formation of NOx. One such strategy, known as rich-quench-lean (RQL), is a method that combusts NH3 in a fuel-rich environment, followed by a quick mix section and a lean burnout section. Rapid mixing of secondary air before lean burnout is thought to be important to minimize the formation of NOx. A genetic algorithm (GA) is used to parametrically vary the secondary air injector design, including the diameter, count, and angles over a constrained design space. The designs are evaluated using a non-reacting OpenFOAM model, with the objective function being the uniformity index of the secondary air (modeled as a scalar). Optimization campaigns show that traditional correlation-based designs might not be adequate. After running 414 OpenFOAM models, the optimal design is 32, 1 mm diameter, angled air injectors, increasing the uniformity index at the outlet of the quick mix zone by 36.8%. The most promising designs will be manufactured and tested in a small RQL burner setup, which is expected to lead to validation and insight into the practical usage of NH3 combustion for power generation.

ammonia combustion↗

DOE OSTI2022

Application of Objectives-Driven Assurance Cases to System Development in an Evolving Acquisition Model

System properties such as “safety” and “dependability” cannot, in practice, be proven, and must be argued in an “assurance case” aimed at supporting risk-acceptance decisions that have to be made by system acquirers and/or regulatory authorities. The paper is concerned with applications of the “assurance case” idea early in design and development of new systems, when (apart from dedicated testing) the only available operating experience information derives from previous (non-identical) systems. Much of the discussion is based on an evolving acquisition model at the US National Aeronautics and Space Administration; previously, most major systems were developed in-house, but some major systems will now be developed by and acquired from commercial providers. Key points discussed include the following. (1) By promoting a particular kind of focused discussion between acquirers and providers, the use of assurance cases should be particularly valuable under the new acquisition model. (2) In principle, objectives-driven (sometimes called “performance-based”) approaches to assurance of performance have significant advantages in cases where they are applicable. (3) For truly novel systems, completeness of the safety analysis is a significant issue; it is important for the assurance case to include a commitment by the provider (or applicant) to seriously pursue analysis of operating experience, so that previously unrecognized hazards can be identified and addressed. (4) Inquiries into major accidents often point to deficiencies in management oversight in all parts of the life cycle; management processes need to be addressed in the formulation and the implementation of an assurance case. Under the new acquisition model, these considerations imply a serious reconsideration of the way in which the development process is managed by both providers and acquirers.

99 GENERAL AND MISCELLANEOUS↗

NASA NTRS2015

Radiant Heating Tests on Meteoritic Material

A new effort geared toward modeling the physics of meteor entry and break-up is underway at NASA Ames Research Center. This is part of a broader interdisciplinary effort on providing physics-based risk assessment models for potentially hazardous objects. As part of the entry modeling task we are seeking to improve our understanding of, among other things, the ablation of meteoric material during high speed entry into earths atmosphere. Meteoroid entry differs greatly in some key respects from spacecraft entry modeling. First, the aerothermal environment at these high velocities (18 km/s) is dominated by radiation. Second, meteoroids less than, say, 50m in size, will likely lose a significant portion of their mass during the high-speed atmospheric entry process due to vaporization as well as melting and spallation. The mass of the object, in turn, directly affects the amount of energy deposited in the atmosphere, and therefore the amount of damage done. Thus it is important for us to understand and be able to model this process in greater detail in order to assess the hazard posed by these objects.In this presentation, we first give an overview of the simple ablation models that are typically used in meteor entry calculations. Additionally, we will present a new model which utilizes a similar approach to what is typically done for spacecraft TPS response modeling. This uses an equilibrium assumption near the surface to compute the ablation rate, as is done in heritage material response codes. Next we describe the radiant heating experiment which uses a high-powered laser to emulate the radiation dominated heating environment experienced by meteoroids during atmospheric entry. The facility the Laser Hardened Material Evaluation Laboratory (LHMEL) permits us to expose samples of meteoritic material to heating rates in excess of 100kW/sq.cm. An overview of the experimental set-up and test plan for the initial exploratory campaign at this facility will be given. Then we present both qualitative and quantitative results from this initial test series. Comparisons between predicted and measured ablation rates suggest that there may be significant blockage of the incident beam by the ablation plume. Furthermore, melt is shown to be a significant ablation mechanics, even at high heating rates (16 kW/sq.cm). Finally, comparisons between the phenomenology of the ablation of terrestrial rocks namely, basalt -- to that of meteorites show very different behavior. This is shown to likely be due in part to the effect of composition on the melt viscosity.

Stern, Eric↗

DOE OSTI2020

Vadose Zone Model for REDOX Area for Composite Analysis

The objectives of the vadose modeling for the updated Hanford Site composite analysis (CA) are to simulate the flow and transport of water and radionuclide releases from the surface to the water table and to provide radionuclide transfer rates for the plateau to river (P2R) model, version 8.3 (CP-57037, Model Package Report: Plateau to River Groundwater Model, Version 8.3). Water additions include natural recharge and water discharged to the ground as a result of industrial processes associated with Hanford Site operations. Contaminant sources include radionuclides in water discharged to the ground during operations and radionuclides disposed “dry” in solid waste burial grounds or other means. The following 16 radionuclides were selected for this modeling effort: carbon-14 (C-14), chlorine-36 (Cl-36), tritium (H-3), iodine-129 (I-129), neptunium-237 (Np-237), rhenium-187 (Re-187), strontium-90 (Sr-90), technetium-99 (Tc-99), uranium-232 (U-232), uranium-233 (U-233), uranium-234 (U-234), uranium-235 (U-235), uranium-236 (U-236), uranium-238 (U-238), radium-226 (Ra-226), and thorium-230 (Th-230). The simulation time starts in 1943 and ends at 12070, which is 10,000 years after assumed Hanford Site closure in 2070. The parallel version of the Subsurface Transport Over Multiple Phases (STOMP1) simulator, officially named the exascale Subsurface Transport Over Multiple Phases (eSTOMP), is used to simulate flow and transport for the vadose models. The documentation for the STOMP code is comprehensive. The theoretical and numerical approaches applied in the STOMP code are documented in a published theory guide (PNNL-12030, STOMP Subsurface Transport Over Multiple Phases Version 2.0 Theory Guide). The code has undergone a rigorous verification procedure against analytical solutions, laboratory-scale experiments, and field-scale demonstrations. The application guide (PNNL-11216, STOMP Subsurface Transport Over Multiple Phases Application Guide) provides instructive examples in the application of the code to classical groundwater problems. The user’s guide (PNNL-15782, STOMP: Subsurface Transport Over Multiple Phases Version 4.0: User’s Guide) describes the general use, input file formatting, compilation, and execution of the code. The primary output of the vadose zone modeling is radionuclide transfer rates to the groundwater for input into the P2R model. The rates will be summed over the 100 by 100 m P2R grid cells that fall within the vadose zone model source domain.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

Vadose Zone Model for PFP Area for Composite Analysis

The objectives of the vadose modeling for the updated Hanford Site composite analysis (CA) are to simulate the flow and transport of water and radionuclide releases from the surface to the water table and to provide radionuclide transfer rates for the plateau to river (P2R) model, version 8.3 (CP-57037, Model Package Report: Plateau to River Groundwater Model, Version 8.3). Water additions include natural recharge and water discharged to the ground as a result of industrial processes associated with Hanford Site operations. Contaminant sources include radionuclides in water discharged to the ground during operations and radionuclides disposed “dry” in solid waste burial grounds or other means. The following 16 radionuclides were selected for this modeling effort: carbon-14 (C-14), chlorine-36 (Cl-36), tritium (H-3), iodine-129 (I-129), neptunium-237 (Np-237), rhenium-187 (Re-187), strontium-90 (Sr-90), technetium-99 (Tc-99), uranium-232 (U-232), uranium-233 (U-233), uranium-234 (U-234), uranium-235 (U-235), uranium-236 (U-236), uranium-238 (U-238), radium-226 (Ra-226), and thorium-230 (Th-230). The simulation time starts in 1943 and ends at 12070, which is 10,000 years after assumed Hanford Site closure in 2070. The parallel version of the Subsurface Transport Over Multiple Phases (STOMP1) simulator, officially named the exascale Subsurface Transport Over Multiple Phases (eSTOMP), is used to simulate flow and transport for the vadose models. The documentation for the STOMP code is comprehensive. The theoretical and numerical approaches applied in the STOMP code are documented in a published theory guide (PNNL-12030, STOMP Subsurface Transport Over Multiple Phases Version 2.0 Theory Guide). The code has undergone a rigorous verification procedure against analytical solutions, laboratory-scale experiments, and field-scale demonstrations. The application guide (PNNL-11216, STOMP Subsurface Transport Over Multiple Phases Application Guide) provides instructive examples in the application of the code to classical groundwater problems. The user’s guide (PNNL-15782, STOMP: Subsurface Transport Over Multiple Phases Version 4.0: User’s Guide) describes the general use, input file formatting, compilation, and execution of the code. The primary output of the vadose zone modeling is radionuclide transfer rates to the groundwater for input into the P2R model. The rates will be summed over the 100 by 100 m P2R grid cells that fall within the vadose zone model source domain.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

Vadose Zone Model for PUREX Area for Composite Analysis

The objectives of the vadose modeling for the updated Hanford Site composite analysis (CA) are to simulate the flow and transport of water and radionuclide releases from the surface to the water table and to provide radionuclide transfer rates for the plateau to river (P2R) model, version 8.3 (CP-57037, Model Package Report: Plateau to River Groundwater Model, Version 8.3). Water additions include natural recharge and water discharged to the ground as a result of industrial processes associated with Hanford Site operations. Contaminant sources include radionuclides in water discharged to the ground during operations and radionuclides disposed “dry” in solid waste burial grounds or other means. The following 16 radionuclides were selected for this modeling effort; carbon-14 (C-14), chlorine-36 (Cl-36), tritium (H-3), iodine-129 (I-129), neptunium-237 (Np-237), rhenium-187 (Re-187), strontium-90 (Sr-90), technetium-99 (Tc-99), uranium-232 (U-232), uranium-233 (U-233), uranium-234 (U-234), uranium-235 (U-235), uranium-236 (U-236), uranium-238 (U-238), radium-226 (Ra-226), and thorium-230 (Th-230). The simulation time starts in 1943 and ends at 12070, which is 10,000 years after assumed Hanford Site closure in 2070.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

Vadose Zone Model for B Complex for Composite Analysis

The objectives of the vadose modeling for the updated Hanford Site composite analysis (CA) are to simulate the flow and transport of water and radionuclide releases from the surface to the water table and to provide radionuclide transfer rates for the plateau to river (P2R) model, version 8.3. Water additions include natural recharge and water discharged to the ground as a result of industrial processes associated with Hanford Site operations. Contaminant sources include radionuclides in water discharged to the ground during operations and radionuclides disposed “dry” in solid waste burial grounds or other means. The simulation time starts in 1943 and ends at 12070, which is 10,000 years after assumed Hanford Site closure in 2070. The scope of this ECF is to document the development and results of the B Complex vadose zone model. The B Complex model encompasses the region around single-shell tank farms 241-B, 241-BX, and 241-BY, and adjacent cribs, trenches, and burial grounds in the northwestern part of the 200 East Area. Sources of waste to this area were primarily from B Plant (221-B) and U Plant (221-U).

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

NASA NTRS1985

Constitutive modeling for isotropic materials

The objective is to develop a unified constitutive model for finite element structural analysis of turbine engine hot-section components. This effort constitutes a different approach for non-linear finite-element computer codes which have heretofore been based on classical inelastic methods. The unified constitutive theory to be developed will avoid the simplifying assumptions of classical theory and should more accurately represent the behavior of superalloy materials under cyclic loading conditions and high temperature environments. During the first two years of the program, extensive experimental correlations were made with two representative unified models. The experiments were both uniaxial and biaxial at temperatures up to 1093 C (2000 F). In addition, the unified models were adopted to the MARC finite element code and used for stress analysis of notched bar and turbine blade geometries.

Lindholm, Ulric S.↗

NASA NTRS2012

Model-Based Fault Diagnosis: Performing Root Cause and Impact Analyses in Real Time

Generic, object-oriented fault models, built according to causal-directed graph theory, have been integrated into an overall software architecture dedicated to monitoring and predicting the health of mission- critical systems. Processing over the generic fault models is triggered by event detection logic that is defined according to the specific functional requirements of the system and its components. Once triggered, the fault models provide an automated way for performing both upstream root cause analysis (RCA), and for predicting downstream effects or impact analysis. The methodology has been applied to integrated system health management (ISHM) implementations at NASA SSC's Rocket Engine Test Stands (RETS).

Figueroa, Jorge F.↗

DOE OSTI2019

Comprehensive analysis of technical efficiency of business establishments in Santa Barbara County: Evaluation of spatial and aspatial stochastic frontier models

One of the main objectives of land-use and transport models is to accurately describe the organization and composition of the regional economy. However, these models currently lack the integration of competitiveness and performance of the regional economy, which in-turn is dependent on performance and efficiency of business establishments in the region. In this study, we present a framework that integrates the aspects of economic efficiency of business establishments within the context of land-use and transport models. For this purpose, we comprehensively assess the impacts of both internal and external attributes of business establishment on its efficiency. In this process, we specifically focus on aspects of incorporating locational characteristics using two frameworks - (a) measuring locational attributes using link-based multiple centrality indicators (L-MCI) and (b) measuring neighborhood effects of business establishments through spatial proximity that account for co-location and competition. We make use of the stochastic frontier modeling framework to investigate the factors impacting business establishment-level output measured as sales per unit area. Specifically, the aspatial stochastic frontier (ASF) model incorporates L-MCI measures as covariates and the spatial stochastic frontier (SSF) model incorporates neighborhood effects of business establishments through network distance measured using the roadway network for Santa Barbara County.Overall, the empirical results suggest that there exist significant inefficiencies across all business establishment types that are mainly impacted by internal attributes of the business. For example, larger and older business establishments are more efficient than business establishments that are smaller in size and younger in age. Furthermore, the results also suggest that locational impacts are better captured using the ASF specification when compared with SSF specification. For instance, the spatial correlation estimate (from SSF) although statistically significant, is very weak. Thereby, indicating a lack of spatial structure in technical efficiency of businesses manifested in the form of co-location and competition. In contrast, the ASF model captures spatial structure of technical efficiency thoroughly with inclusion of the L-MCI measures. This indicates that L-MCI measures have varying levels of impact on efficiency (positive or negative) depending on its relative location in the region. For instance, retail trade businesses have higher levels of revenue (or sales per unit area) when located along links in a grid-network like structure e.g. downtown areas. Additionally, businesses in professional services have no impact on their efficiency based on the configuration and structure of the regional transportation network. However, results from both specifications indicate that there exist major benefits on efficiency of business establishments from local competition for retail trade and healthcare services.

Ravulaparthy, Srinath↗

NASA NTRS1993

Fusing modeling techniques to support domain analysis for reuse opportunities identification

Functional modeling techniques or object-oriented graphical representations, which are more useful to someone trying to understand the general design or high level requirements of a system? For a recent domain analysis effort, the answer was a fusion of popular modeling techniques of both types. By using both functional and object-oriented techniques, the analysts involved were able to lean on their experience in function oriented software development, while taking advantage of the descriptive power available in object oriented models. In addition, a base of familiar modeling methods permitted the group of mostly new domain analysts to learn the details of the domain analysis process while producing a quality product. This paper describes the background of this project and then provides a high level definition of domain analysis. The majority of this paper focuses on the modeling method developed and utilized during this analysis effort.

Hall, Susan Main↗

DOE OSTI2025

Multi-objective surrogate-assisted calibration of CPFEM models using macroscopic response and in situ EBSD measurements of grain reorientation trajectories

Crystal plasticity finite element method (CPFEM) models are widely used to simulate the deformation behaviour of polycrystalline materials, but their calibration is often limited by their high computational cost and the non-convexity of the optimisation landscape. Here, this study develops a multi-objective surrogate-assisted calibration workflow that couples a multi-objective genetic algorithm (MOGA) with an adaptively trained deep neural network (DNN) surrogate model to efficiently identify CPFEM parameters from experimental data. The workflow is demonstrated on three crystal plasticity (CP) formulations of increasing complexity — Voce hardening (VH), two-coefficient latent hardening (LH2), and six-coefficient latent hardening (LH6) — using in situ electron backscatter diffraction (EBSD) measurements of Alloy 617 under uniaxial tensile loading. The CPFEM models are calibrated against the experimentally observed stress–strain response and reorientation trajectories of eight grains, then validated against eight additional trajectories and overall texture evolution. Across the CP formulations, the macroscopic response was reproduced reliably, while differences emerged in the robustness and accuracy of the grain-scale predictions. Including grain reorientation trajectories in the multi-objective calibration improved texture evolution predictions and filtered out physically inconsistent parameter sets that can arise from calibrating against only the stress–strain data. The workflow also demonstrates good transferability of calibrated parameters from a low- to a high-fidelity microstructural model. These results provide practical guidance for integrating in situ microstructural data into CPFEM through efficient, repeatable, and physically meaningful multi-objective calibration.

Crystal plasticity finite element method↗