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A practical solution algorithm for steady 3D Euler flows is presented. This algorithm employs coupling of a surface triangulator, an automatic tetrahedral mesh generator, an unstructured grid flow solver, and an error estimation procedure. The performance of the method is illustrated using a shock interaction problem in high Mach number flow over a swept circular cylinder.
Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.
Thermal radiation transport is integral to multiphysics applications such as inertial confinement fusion and astrophysics. Materials in these problems can be exceptionally optically thick and diffusive. A transport solver must be capable of accurate and efficient solutions to such problems. Previously, Woods et al. established a proof-of-concept for high order (HO) discontinuous Galerkin finite element (DGFEM) transport on meshes with curved surfaces. They numerically demonstrated the spatial convergence is order p + 1. Remarkably, these convergence rates are relatively invariant to the amount of curvature of the spatial mesh. We present here an extension of this work that demonstrates a rapidly-convergent iterative solver for the FEM transport equations on meshes with curved sides. We use the modified interior penalty (MIP) diffusion synthetic acceleration (DSA) established by Wang and Ragusa. The partially consistent MIP equations stem from the symmetric interior penalty (IP) derivation of the diffusion equation. The IP method is not stable for optically thick cells so it was combined with the diffusion conforming form (DCF), a discretized diffusion equation derived from the discretized S N equations that is not stable for intermediate optical thicknesses. The MIP equations adapt a 'switch' to the IP method in optically thinner regions and the DCF method in optically thicker regions. The literature does not reveal any Fourier analyses performed for iterative transport on meshes with curved surfaces. Wang and Ragusa performed a Fourier analysis for orthogonal meshes and our results appear to be as expected. Previous literature has reported on some of the sensitivities that affect the spectral radius of DSA schemes: polynomial order, and scattering ratios. Presently, we focus on the sensitivity of the spectral radius to polynomial order, scattering ratio, the constant C, and mesh curvature for varying cell thicknesses. We also numerically study the behavior of the spectral radius to verify our DSA implementation. A sequence of problems is designed to exercise the acceleration. As the problems become more optically thick and diffusive, the accelerated solution continues to converge toward the analytic solution. We employ the Modular Finite Element Methods (MFEM) package, developed at Lawrence Livermore National Laboratory, to generate the linear algebraic system of equations for each angle in the discrete ordinates quadrature set, and use UMFPack to solve this system of equations.
The concrete biological shield in light-water reactors is exposed to neutron and gamma irradiation, which deteriorates the concrete’s mechanical properties in the long term. To assess the irradiation-induced damage, predictive mechanical models are developed and used in parallel with the characterization of irradiated concrete samples. Realistic 3D simulation domains can drastically improve a model’s prediction. In this work, we utilized x-ray computed tomography (XCT) data of a concrete specimen to reconstruct its 3D microstructure. The XCT data shows low contrast between the concrete’s aggregates and cement paste, resulting in poor image segmentation when using traditional unsupervised techniques. To address this issue, we developed and trained a 2.5D U-Net model on only 24 pre-labeled XCT layers to segment 651 layers of the XCT data. The overall F1-score of the model is approximately 96%. Then, we created a 3D finite element (FE) mesh based on the stack of segmented images. The FE model contains radiation-induced expansion, damage, and creep. The constitutive equations are adapted to each phase (aggregates and cement paste). Here, we simulated the effects of neutron irradiation in the concrete specimen as well as the specimen’s mechanical response to uniaxial compression. Finally, model validation was performed using experimental data on similar concrete specimens in the literature.
Here, we study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element-agglomeration-based algebraic multigrid (AMGe) technique with improved approximation properties is used, for the first time, to generate upscaled and accurate coarse systems for the reservoir simulation equations. The upscaling technique is applied to both the mixed system for velocity and pressure and to the hyperbolic transport equations, providing fully upscaled systems. By introducing additional degrees of freedom associated with nonplanar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart--Thomas and piecewise discontinuous polynomial spaces, resulting in strong mass conservation for the upscaled systems. Due to the guaranteed approximation properties and the generic nature of the AMGe method, recursive multilevel upscaling is automatically obtained. Furthermore, this technique works for both structured and unstructured meshes. Multiscale mixed finite elements exhibit accuracy for general unstructured meshes but do not in general lead to nested hierarchy of spaces. Multiscale multilevel mimetic finite differences generate nested spaces but lack the adaptivity of the flux representation on coarser levels that the proposed AMGe approach offers. Thus, the proposed approach can be seen as a rigorous bridge that merges the best properties of these two existing methods. The accuracy and stability of the studied multilevel AMGe upscaling technique is demonstrated on two challenging test cases.
This application can determine the performance and chemical behavior of batteries in 1D when they are cycled. With CanTrilBat, we are developing predictive phenomenological models for battery systems to predict operating performance and rate limiting steps in the performance of battery models. Particular attention is paid to primary and secondary chemistry mechanisms, such as the thermal runaway mechanisms experienced in secondary lithium ion batteries or self-discharge reaction mechanism that all batteries experience to one extent or another. The first application of this model has been for modeling the performance of thermal batteries. However, an implementation for secondary ion batteries is next. CanTrilBat applications solves transient problems involving batteries. It is a 1-D application that represents 3-D physical systems that can be reduced using the porous flow approximation for the anode, cathode, and separator. A control volume formulation is used to track conserved quantities. An operator-split approach is used to calculate the chemistry, diffusion and electronic transport that occurs within cathode and anode particles, allowing for the reduction in code complexity. All jacobian operations in CanTrilBat utilize numerical jacobians. A home grown predictor corrector scheme is used for time step control, and a home grown newton solver is used to relax the equations at each time step. Trilinos is used to solve for the resulting linear systems equation A block text input format is used to initialize options for the CanTriBat program. Within this block are the names of several XML input files, which specify the chemistry mechanism. These XML input files are read by low level Cantera routines, and serve to initialize the electrode and electrolyte chemistry mechanisms and transport properties. A GUI implementation has been contracted out to a university professor, but has not been implemented yet. It?s expected that CanTriBat will have the capability to do adaptive grid refinement in later versions. Grids will be halved based on an complicated set of heuristic criteria, based on how well the mesh captures changes to the first and second derivatives of selected solution components. Primary emphasis is given towards converged profiles for IV performance.
We studied the interaction of a vortical density inhomogeneity (VDI) with shock and expansion waves. We call the VDI the region of concentrated vorticity (vortex) with a density different from that of ambiance. Non-parallel directions of the density gradient normal to the VDI surface and the pressure gradient across a shock wave results in an additional vorticity. The roll-up of the initial round VDI towards a non-symmetrical shape is studied numerically. Numerical modeling of this interaction is performed by a 2-D Euler code. The use of an adaptive unstructured numerical grid makes it possible to obtain high accuracy and capture regions of induced vorticity with a moderate overall number of mesh points. For the validation of the code, the computational results are compared with available experimental results and good agreement is obtained. The interaction of the VDI with a propagating shock wave is studied for a range of initial and induced circulations and obtained flow patterns are presented. The splitting of the VDI develops into the formation of a non-symmetrical vortex pair and not in a set of vortices. A method for the analytical computation of an overall induced circulation Gamma(sub 1) as a result of the interaction of a moving VDI with a number of waves is proposed. Simplified, approximated, expressions for Gamma(sub 1) are derived and their accuracy is discussed. The splitting of the VDI passing through the Prandtl-Meyer expansion wave is studied numerically. The obtained VDI patterns are compared to those for the interaction of the VDI with a propagating shock wave for the same values of initial and induced circulations. These patterns have similar shapes for corresponding time moments.
This application can determine the performance and chemical behavior of batteries in 1D when they are cycled. With CanTrilBat, we are developing predictive phenomenological models for battery systems to predict operating performance and rate limiting steps in the performance of battery models. Particular attention is paid to primary and secondary chemistry mechanisms, such as the thermal runaway mechanisms experienced in secondary lithium ion batteries or self-discharge reaction mechanism that all batteries experience to one extent or another. The first application of this model has been for modeling the performance of thermal batteries. However, an implementation for secondary ion batteries is next. CanTrilBat applications solves transient problems involving batteries. It is a 1-D application that represents 3-D physical systems that can be reduced using the porous flow approximation for the anode, cathode, and separator. A control volume formulation is used to track conserved quantities. An operator-split approach is used to calculate the chemistry, diffusion and electronic transport that occurs within cathode and anode particles, allowing for the reduction in code complexity. All jacobian operations in CanTrilBat utilize numerical jacobians. A home grown predictor corrector scheme is used for time step control, and a home grown newton solver is used to relax the equations at each time step. Trilinos is used to solve for the resulting linear systems equation A block text input format is used to initialize options for the CanTriBat program. Within this block are the names of several XML input files, which specify the chemistry mechanism. These XML input files are read by low level Cantera routines, and serve to initialize the electrode and electrolyte chemistry mechanisms and transport properties. A GUI implementation has been contracted out to a university professor, but has not been implemented yet. It?s expected that CanTriBat will have the capability to do adaptive grid refinement in later versions. Grids will be halved based on an complicated set of heuristic criteria, based on how well the mesh captures changes to the first and second derivatives of selected solution components. Primary emphasis is given towards converged profiles for IV performance.
A finite-element remeshing approach that makes use of quadrilateral and triangular elements is described. The approach uses the solution on a previous mesh to create a new mesh. Meshes are completely unstructured with highly refined elements in regions of steep gradients and larger elements where gradients are smaller. Studies of convergence rates for heat conduction problems with exact solutions show that for problems with highly localized solution variations, the remeshing approach gives smaller solution errors with fewer unknowns than refinement of uniform, structured meshes.
Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving physical systems described by partial differential equations (PDEs). However, their accuracy in dynamical systems, particularly those involving sharp moving boundaries with complex initial morphologies, remains a challenge. Here, this study introduces an approach combining residual-based adaptive refinement (RBAR) with causality-informed training to enhance the performance of PINNs in solving spatio-temporal PDEs. Our method employs a three-step iterative process: initial causality-based training, RBAR-guided domain refinement, and subsequent causality training on the refined mesh. Applied to the Allen-Cahn equation, a widely-used model in phase field simulations, our approach demonstrates significant improvements in solution accuracy and computational efficiency over traditional PINNs. Notably, we observe an ‘overshoot and relocate’ phenomenon in dynamic cases with complex morphologies, showcasing the method’s adaptive error correction capabilities. This synergistic interaction between RBAR and causality training enables accurate capture of interface evolution, even in challenging scenarios where traditional PINNs fail. Our framework not only resolves the limitations of uniform refinement strategies but also provides a generalizable methodology for solving a broad range of spatio-temporal PDEs. The enhanced performance of the RBAR–causality combined framework demonstrates its strong potential for advancing PINN-based modeling of physical systems characterized by complex, evolving interfaces.
An interactive adaptive remeshing algorithm utilizing a frontal grid generator and a variety of time integration schemes for the two-dimensional Euler equations on unstructured meshes is presented. Several device dependent interactive graphics interfaces have been developed along with a device independent DI-3000 interface which can be employed on any computer that has the supporting software including the Cray-2 supercomputers Voyager and Navier. The time integration methods available include: an explicit four stage Runge-Kutta and a fully implicit LU decomposition. A cell-centered finite volume upwind scheme utilizing Roe's approximate Riemann solver is developed. To obtain higher order accurate results a monotone linear reconstruction procedure proposed by Barth is utilized. Results for flow over a transonic circular arc and flow through a supersonic nozzle are examined.
A new methodology is developed to simulate unsteady flows about prescribed and aerodynamically determined moving boundary problems. The method couples the fluid dynamics and rigid-body dynamics equations to capture the time-dependent interference between stationary and moving boundaries. The unsteady, compressible, inviscid (Euler) equations are solved on dynamic, unstructured grids by an explicit, finite-volume, upwind method. For efficiency, the grid adaptation is performed within a window around the moving object. The Eulerian equations of the rigid-body dynamics are solved by a Runge-Kutta method in a non-inertial frame of reference. The two-dimensional flow solver is validated by computing the flow past a sinusoidally-pitching airfoil and comparing these results with the experimental data. The overall methodology is used for two two-dimensional examples: the flow past an airfoil which is performing a three-degrees-of-freedom motion in a transonic freestream, and the free-fall of a store after separation from a wing-section. Then the unstructured mesh methodology is extended to three-dimensions to simulate unsteady flow past bodies in relative motion, where the trajectory is determined from the instantaneous aerodynamics. The flow solver and the adaptation scheme in three dimensions are validated by simulating the transonic, unsteady flow around a wing undergoing a forced, periodic, pitching motion, and comparing the results with the experimental data. To validate the trajectory code, the six-degrees-of-freedom motion of a store separating from a wing was computed using the experimentally determined force and moment fields, then comparing with an independently generated trajectory. Finally, the overall methodology was demonstrated by simulating the unsteady flowfield and the trajectory of a store dropped from a wing. The methodology, its computational cost notwithstanding, has proven to be accurate, automated, easy for dynamic gridding, and relatively efficient for the required man-hours.
The multi-dimensional self-adaptive grid code, SAGE, is an important tool in the field of computational fluid dynamics (CFD). It provides an efficient method to improve the accuracy of flow solutions while simultaneously reducing computer processing time. Briefly, SAGE enhances an initial computational grid by redistributing the mesh points into more appropriate locations. The movement of these points is driven by an equal-error-distribution algorithm that utilizes the relationship between high flow gradients and excessive solution errors. The method also provides a balance between clustering points in the high gradient regions and maintaining the smoothness and continuity of the adapted grid, The latest version, Version 3, includes the ability to change the boundaries of a given grid to more efficiently enclose flow structures and provides alternative redistribution algorithms.
An upwind cell-centered scheme for the solution of steady laminar viscous high-speed flows is implemented on unstructured two-dimensional meshes. The first-order implementation employs Roe's (1981) approximate Riemann solver, and a higher-order extension is produced by using linear reconstruction with limiting. The procedure is applied to the solution of inviscid subsonic flow over an airfoil, inviscid supersonic flow past a cylinder, and viscous hypersonic flow past a double ellipse. A detailed study is then made of a hypersonic laminar viscous flow on a 24-deg compression corner. It is shown that good agreement is achieved with previous predictions using finite-difference and finite-volume schemes. However, these predictions do not agree with experimental observations. With refinement of the structured grid at the leading edge, good agreement with experimental observations for the distributions of wall pressure, heating rate and skin friction is obtained.
New manufacturing technologies such as additive manufacturing require research and development to minimize the uncertainties in the produced parts. The research involves experimental measurements and large simulations, which result in huge quantities of data to store and analyze. We address this challenge by alleviating the data storage requirements using lossy data compression. We select wavelet bases as the mathematical tool for compression. Unlike images, additive manufacturing data is often represented on irregular geometries and unstructured meshes. Thus, we use Alpert tree-wavelets as bases for our data compression method. We first analyze different basis functions for the wavelets and find the one that results in maximal compression and miminal error in the reconstructed data. We then devise a new adaptive thresholding method that is data-agnostic and allows a priori estimation of the reconstruction error. Finally, we propose metrics to quantify the global and local errors in the reconstructed data. One of the error metrics addresses the preservation of physical constraints in reconstructed data fields, such as divergence-free stress field in structural simulations. While our compression and decompression method is general, we apply it to both experimental and computational data obtained from measurements and thermal/structural modeling of the sintering of a hollow cylinder from metal powders using a Laser Engineered Net Shape process. The results show that monomials achieve optimal compression performance when used as wavelet bases. The new thresholding method results in compression ratios that are two to seven times larger than the ones obtained with commonly used thresholds. Overall, adaptive Alpert tree-wavelets can achieve compression ratios between one and three orders of magnitude depending on the features in the data that are required to preserve. Furthermore, these results show that Alpert tree-wavelet compression is a viable and promising technique to reduce the size of large data structures found in both experiments and simulations.
Low-mass “dwarf” galaxies represent the most significant challenges to the cold dark matter (CDM) model of cosmological structure formation. Because these faint galaxies are (best) observed within the Local Group (LG) of the Milky Way (MW) and Andromeda (M31), understanding their formation in such an environment is critical. We present first results from the Latte Project: the Milky Way on Feedback in Realistic Environments (FIRE). This simulation models the formation of an MW-mass galaxy to z=0 within ΛCDM cosmology, including dark matter, gas, and stars at unprecedented resolution: baryon particle mass of 7070 M{sub ⊙} with gas kernel/softening that adapts down to 1 pc (with a median of 25--60 pc at z=0). Latte was simulated using the GIZMO code with a mesh-free method for accurate hydrodynamics and the FIRE-2 model for star formation and explicit feedback within a multi-phase interstellar medium. For the first time, Latte self-consistently resolves the spatial scales corresponding to half-light radii of dwarf galaxies that form around an MW-mass host down to M{sub star}≳10{sup 5} M{sub ⊙}. Latte’s population of dwarf galaxies agrees with the LG across a broad range of properties: (1) distributions of stellar masses and stellar velocity dispersions (dynamical masses), including their joint relation; (2) the mass–metallicity relation; and (3) diverse range of star formation histories, including their mass dependence. Thus, Latte produces a realistic population of dwarf galaxies at M{sub star}≳10{sup 5} M{sub ⊙} that does not suffer from the “missing satellites” or “too big to fail” problems of small-scale structure formation. We conclude that baryonic physics can reconcile observed dwarf galaxies with standard ΛCDM cosmology.
Recent advances in computational fluid dynamics (CFD) are discussed in reviews and reports. Topics addressed include CFD models in plasma dynamics, parallel computation for simulation studies, CFD for hypersonic airbreathing aircraft, multigrid methods for the steady incompressible Navier-Stokes equations, upwind differencing techniques, TV stable schemes for shock-interacting flows, Euler models of hypersonic vortex flows, parallel multilevel adaptive methods, and vortex methods for slightly viscous three-dimensional flows. Consideration is given to the accuracy of node-based solutions on irregular meshes, multigrid calculations for cascades, a finite-volume-element method for planar cavity flow, parallel heterogeneous mesh refinement for advection-diffusion equations, the convergence of the spectral-viscosity method for nonlinear conservation laws, and numerical simulations of Taylor vortices in a spherical gap.
Purpose: To generate virtual phantoms with clinically relevant deformation and use them to objectively evaluate geometric and dosimetric uncertainties of deformable image registration (DIR) algorithms. Methods: Ten lung cancer patients undergoing adaptive 3DCRT planning were selected. For each patient, a pair of planning CT (pCT) and replanning CT (rCT) were used as the basis for virtual phantom generation. Manually adjusted meshes were created for selected ROIs (e.g. PTV, lungs, spinal cord, esophagus, and heart) on pCT and rCT. The mesh vertices were input into a thin-plate spline algorithm to generate a reference displacement vector field (DVF). The reference DVF was used to deform pCT to generate a simulated replanning CT (srCT) that was closely matched to rCT. Three DIR algorithms (Demons, B-Spline, and intensity-based) were applied to these ten virtual phantoms. The images, ROIs, and doses were mapped from pCT to srCT using the DVFs computed by these three DIRs and compared to those mapped using the reference DVF. Results: The average Dice coefficients for selected ROIs were from 0.85 to 0.96 for Demons, from 0.86 to 0.97 for intensity-based, and from 0.76 to 0.95 for B-Spline. The average Hausdorff distances for selected ROIs were from 2.2 to 5.4 mm for Demons, from 2.3 to 6.8 mm for intensity-based, and from 2.4 to 11.4 mm for B-Spline. The average absolute dose errors for selected ROIs were from 0.2 to 0.6 Gy for Demons, from 0.1 to 0.5 Gy for intensity-based, and from 0.5 to 1.5 Gy for B-Spline. Conclusion: Virtual phantoms were modeled after patients with lung cancer and were clinically relevant for adaptive radiotherapy treatment replanning. Virtual phantoms with known DVFs serve as references and can provide a fair comparison when evaluating different DIRs. Demons and intensity-based DIRs were shown to have smaller geometric and dosimetric uncertainties than B-Spline. Z Shen: None; K Bzdusek: an employee of Philips Healthcare; J Greskovich: None; P Xia: received research grants from Philips Healthcare and Siemens Healthcare.