Search NASA⌕ Search

SEARCH · Search NASA

Results for “weak solutions”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records

Global weak solutions for a nonlocal multispecies Fokker–Planck–Landau system

The global-in-time existence of weak solutions to a spatially homogeneous multispecies Fokker–Planck–Landau system for plasmas in the three-dimensional whole space is shown. The Fokker–Planck–Landau system is a simplification of the Landau equations assuming a linearized, velocity-independent, and isotropic kernel. The resulting equations depend nonlocally and nonlinearly on the moments of the distribution functions via the multispecies local Maxwellians. Furthermore, the existence proof is based on a three-level approximation scheme, energy and entropy estimates, as well as compactness results, and it holds for both soft and hard potentials.

97 MATHEMATICS AND COMPUTING↗

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L 2 convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H -1 -type norm, but not L 2 -coercive. Nevertheless, the L 2 convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L 2 norm, or by properly choosing the finite element spaces, providing implicit control of the L 2 norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L 2 convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. Finally, we believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

97 MATHEMATICS AND COMPUTING↗

The eXtended virtual element method for elliptic problems with weakly singular solutions

This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The X-VEM is formulated by incorporating appropriate enrichment functions in the local spaces. The method is designed to handle highly generic enrichment functions, including singularities arising from fractured domains. By achieving consistency on the enrichment space, the method is proven to achieve arbitrary approximation orders even in the presence of singular solutions. The paper includes a complete convergence analysis under general assumptions on mesh regularity, and numerical experiments validating the method’s accuracy on various mesh families, demonstrating optimal convergence rates in the L 2 - and H 1 - norms on fractured or L-shaped domains.

97 MATHEMATICS AND COMPUTING↗

Qualitative properties of mathematical model for data flow

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

97 MATHEMATICS AND COMPUTING↗

On the continuity of solutions to anisotropic elliptic operators in the limiting case

Abstract We show that local weak solutions to anisotropic elliptic equations with bounded and measurable coefficients, whose prototype is are continuous. Our analysis exploits a particular intrinsic geometry that, within the limiting case on the exponents, allows for a reduction of the oscillation of the solution.

Ciani, Simone [Department of Mathematics, Universi↗

On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system

We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier–Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.

Weber, Franziska↗

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. In this work, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

97 MATHEMATICS AND COMPUTING↗

Weakly quasisymmetric near-axis solutions to all orders

Here, we show that the equations satisfied by weakly quasisymmetric magnetic fields can be solved to arbitrarily high order in powers of the distance from the magnetic axis. This demonstration does not consider force balance. The existence of solutions requires an appropriate choice of parameters, most notably the toroidal current or rotational transform profiles. We do not prove that the expansion converges (it is likely divergent but asymptotic), and thus, the demonstration here should not be taken as definitive proof of the existence of global solutions. Instead, we provide a systematic construction of solutions to an arbitrarily high order.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Midfidelity model verification for a point-absorbing wave energy converter with linear power take-off

In the preliminary design stage of a wave energy converter (WEC), researchers need fast and reliable simulation tools. High-fidelity numerical models are usually employed to study the wave-structure interaction, but the computational cost is demanding. As an alternative, midfidelity models can provide simulations in the order of real time. In this study, we operate Uppsala University’s WEC in a relatively mild sea state and model it using WEC-Sim. The model is verified based on OpenFOAM simulations. To analyze the ability of the midfidelity model to capture WEC dynamics, we investigate the system separately with 1, 2, and 3 degrees of freedom. We examine the contribution of viscous phenomena, and study both linear and weakly nonlinear solutions provided by WEC-Sim. Our results indicate that the viscous effects can be neglected in heave and surge motion, but not for pitch. We also find that the weakly nonlinear WEC-Sim solution successfully agrees with the computational fluid dynamics, whereas the linear solution could suggest misleading results.

16 TIDAL AND WAVE POWER↗

Mechanism of the cadherin–catenin F-actin catch bond interaction

Mechanotransduction at cell–cell adhesions is crucial for the structural integrity, organization, and morphogenesis of epithelia. At cell–cell junctions, ternary E-cadherin/β-catenin/αE-catenin complexes sense and transmit mechanical load by binding to F-actin. The interaction with F-actin, described as a two-state catch bond, is weak in solution but is strengthened by applied force due to force-dependent transitions between weak and strong actin-binding states. Here, we provide direct evidence from optical trapping experiments that the catch bond property principally resides in the αE-catenin actin-binding domain (ABD). Consistent with our previously proposed model, the deletion of the first helix of the five-helix ABD bundle enables stable interactions with F-actin under minimal load that are well described by a single-state slip bond, even when αE-catenin is complexed with β-catenin and E-cadherin. Our data argue for a conserved catch bond mechanism for adhesion proteins with structurally similar ABDs. We also demonstrate that a stably bound ABD strengthens load-dependent binding interactions between a neighboring complex and F-actin, but the presence of the other αE-catenin domains weakens this effect. These results provide mechanistic insight to the cooperative binding of the cadherin–catenin complex to F-actin, which regulate dynamic cytoskeletal linkages in epithelial tissues.

59 BASIC BIOLOGICAL SCIENCES↗

Tuning Magnetic Symmetry and Properties in the Olivine Series Li 1–x Fe x Mn 1–x PO 4 through Selective Delithiation

In this work, we present the first in-depth study of the nuclear and magnetic structures of the fully lithiated LiFe x Mn 1–x PO 4 solid solutions as well as the partially delithiated Li 1– xFe x Mn 1–x PO 4 series using neutron powder diffraction. By using a weak oxidizing solution that only oxidizes Fe 2+ , we find that lithium can be removed from these olivines in a stepwise manner. Representational analysis was performed on the observed magnetic reflections and found that the lithiated solid solutions order in the magnetic space group Pnma', which is identical to triphylite (LiFePO 4 ). We report that, even when removing 20% of the lithium from the hcp lattice, the magnetic structure changes and multiple magnetic contributions are needed to fully encompass the observed magnetic order. We find that the partially delithiated x = 0.2 and 0.3 have a magnetic structure P2 1 '/n11, and when x surpasses 0.3, the magnetic structure changes to P2 1 2 1 2 1 . We give a detailed report on the effect of delithiation on magnetic order, bond distances/angles, and the super- and supersuperexchange interactions in the material. The symmetries of the magnetic order of these solid solutions are then discussed with regard to their potential as ferrotoroidic candidates.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Modeling the Solvation and Acidity of Carboxylic Acids Using an Ab Initio Deep Neural Network Potential

Formic and acetic acid constitute the simplest of carboxylic acids, yet they exhibit fascinating chemistry in the condensed phase such as proton transfer and dimerization. The go-to method of choice for modeling these rare events have been accurate but expensive ab-initio molecular dynamics simulations. Here, we present a deep neural network potential trained using accurate ab-initio data that can be used in tandem with enhanced-sampling methods to perform an efficient exploration of the free-energy surface of aqueous solutions of weak carboxylic acids. In particular, we show that our model captures proton dissociation and provides a good estimate of the pK a , as well as the dimerization of formic and acetic acid. This provides a suitable starting point for applications in different research areas where computational efficiency coupled with the accuracy of ab-initio methods is required.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Canalization of Phenotypes—When the Transcriptome is Constantly but Weakly Perturbed

Abstract Recent studies have increasingly pointed to microRNAs (miRNAs) as the agent of gene regulatory network (GRN) stabilization as well as developmental canalization against constant but small environmental perturbations. To analyze mild perturbations, we construct a Dicer-1 knockdown line (dcr-1 KD) in Drosophila that modestly reduces all miRNAs by, on average, ∼20%. The defining characteristic of stabilizers is that, when their capacity is compromised, GRNs do not change their short-term behaviors. Indeed, even with such broad reductions across all miRNAs, the changes in the transcriptome are very modest during development in stable environment. By comparison, broad knockdowns of other regulatory genes (esp. transcription factors) by the same method should lead to drastic changes in the GRNs. The consequence of destabilization may thus be in long-term development as postulated by the theory of canalization. Flies with modest miRNA reductions may gradually deviate from the developmental norm, resulting in late-stage failures such as shortened longevity. In the optimal culture condition, the survival to adulthood is indeed normal in the dcr-1 KD line but, importantly, adult longevity is reduced by ∼90%. When flies are stressed by high temperature, dcr-1 KD induces lethality earlier in late pupation and, as the perturbations are shifted earlier, the affected stages are shifted correspondingly. Hence, in late stages of development with deviations piling up, GRN would be increasingly in need of stabilization. In conclusion, miRNAs appear to be a solution to weak but constant environmental perturbations.

Lu, Guang-An↗

Graphene nanoplatelets reinforced cement as a solution to leaky wellbores reinforcing weak points in hydrated Portland cement with graphene nanoparticles improves mechanical and chemical durability of wellbore cements

In this work, to improve the performance of wellbore cement under subsurface environments, graphene nanoplatelets (GNPs) were added in various percentages to Class-H cement slurry. Microstructural characterization of cement slurries cured at 90 °C and 95% RH indicates that GNP modifies the microstructure of hydrated cement by reinforcing pore spaces. As a result, the mechanical properties, such as Young's modulus and axial peak stress, obtained from high temperature triaxial compression tests are significantly improved based on different percentages of added GNPs. Furthermore, the hydrated 1 × 2 inch GNP- Portland cement cores tested under simulated deep wellbore conditions of high-temperature and high pressure (HTHP), appear to have a ductile-like behavior, when compared to a typical brittle nature of Portland cement pastes. From our observations and published data on graphene resistance to fracture, GNP addition to cement is evidently enhancing the flexibility of cement. These improvements would reduce the risks associated with wellbore cement deterioration and a consequent leakage in fossil fuel production, geothermal energy production, CO 2 storage as well as long-term sealing materials in plugging and abandonment of all wellbores at the end of their service life.

36 MATERIALS SCIENCE↗

Adaptive Interface-PINNs (AdaI-PINNs) for transient diffusion: Applications to forward and inverse problems in heterogeneous media

We model transient diffusion in heterogeneous materials using a novel physics-informed neural networks framework (PINNs) termed Adaptive interface physics-informed neural networks or AdaI-PINNs (Roy et al. arXiv preprint arXiv:2406.04626, 2024). AdaI-PINNs utilize different activation functions with trainable slopes tailored to each material region within the computational domain, allowing for a fully automated and adaptive PINNs approach to model interface problems with strongly and weakly discontinuous solutions. To enhance its performance in highly heterogeneous transient diffusion systems, we prescribe a suite of robust practices, including appropriate non-dimensionalization of equations, a biased sampling method, Glorot initialization, and the hard enforcement of boundary and initial conditions. Here we evaluate the efficacy of the proposed method on several benchmark forward and inverse problems. Comparative studies on one-dimensional and two-dimensional benchmark problems reveal that the modified AdaI-PINNs outperform its unmodified counterpart, achieving root-mean-square errors that are at least two orders of magnitude better in forward problems. For inverse problems, the maximum errors in the approximated diffusion coefficients by modified AdaI-PINNs are four orders of magnitude better than those of the unmodified version. Additionally, modified AdaI-PINNs demonstrate improved stability in problems with large material mismatches.

42 ENGINEERING↗

Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network

In this study, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model involves a mesoscale version of Darcy’s law to capture the flow mechanism in the tissue matrix. Modeling flow and transport processes in the vasculature supplying the healthy and cancerous tissue, one-dimensional (1D) equations are considered. Since the models governing the transport and flow processes are defined together with cell species models on a three-dimensional (3D) domain, we obtain a 3D–1D coupled model.

3D–1D coupled blood flow models↗

Influence of Ether Solvent and Anion Coordination on Electrochemical Behavior in Calcium Battery Electrolytes

The emergence of magnesium and calcium batteries as potential beyond Li ion energy storage technologies has generated significant interest into the fundamental aspects of alkaline earth metal cation coordination in multivalent electrolytes and the impact of coordination on application-critical electrolyte properties such as solubility, transport, and electrochemical stability. Understanding these details in calcium electrolytes is of immediate importance due to recent, unprecedented demonstrations of reversible calcium metal electrodeposition in a limited number of ethereal solvent-based systems. In this work, we provide insight connecting Ca 2+ coordination tendencies to important calcium battery electrolyte properties. Our results demonstrate a clear solvent:Ca 2+ coordination strength trend across a series of cyclic ether and linear glyme solvents that controls the extent of ion association in solutions of “weakly” coordinating salts. We apply understanding gained from these results to rationalize relative anion:Ca 2+ coordination tendencies and attendant Ca 2+ coordination structures using two oxidatively stable anions of particular interest for current battery electrolytes. Armed with this understanding of solvent and anion interactions with Ca 2+ , we demonstrate and interpret differences in electrochemical calcium deposition behavior across several electrolyte exemplars with varying solvent and anion coordination strengths. Our findings demonstrate that solvents exhibiting especially strong coordination to Ca 2+ , such as triglyme, can inhibit reversible calcium deposition despite effective elimination of anion:Ca 2+ coordination while solvents exhibiting more modest coordination strength, such as 1,2-dimethoxyethane, may enable deposition provided anion:Ca 2+ coordination is substantially limited. These results reveal that the strength of coordination of both anion and solvent should be considered in the design of electrolytes for calcium batteries.

25 ENERGY STORAGE↗