Journal of Computational Physics: X最新文献

Mechanochemical processes on surfaces such as the cellular cortex or epithelial sheets, play a key role in determining patterns and shape changes of biological systems. To understand the complex interplay of hydrodynamics and material flows on such active surfaces requires novel numerical tools. Here, we present a finite-element method for an active deformable surface interacting with the surrounding fluids. The underlying model couples surface and bulk hydrodynamics to surface flow of a diffusible species which generates active contractile forces. The method is validated with previous results based on linear stability analysis and shows almost perfect agreement regarding predicted patterning. Away from the linear regime we find rich non-linear behavior, such as the presence of multiple stationary states. We study the formation of a contractile ring on the surface and the corresponding shape changes. Finally, we explore mechanochemical pattern formation on various surface geometries and find that patterning strongly adapts to local surface curvature. The developed method provides a basis to analyze a variety of systems that involve mechanochemical pattern formation on active surfaces interacting with surrounding fluids.

It is well-known in the Smoothed Particle Hydrodynamics (SPH) community that correction in the gradient and Laplacian operators have the potential to drastically increase the accuracy of the method at the expense of computational stability. This paper proposes a stable implementation of such corrections in all derivative operators to the Arbitrary Lagrangian Eulerian incompressible SPH (ALE-ISPH) method, in addition to a novel Neumann boundary condition (BC) applied directly on the velocity (as opposed to traditional BCs where the constraint is applied on the acceleration). In this way, the pressure is solved for both water and wall particles simultaneously, leading to a pressure field that obeys non-penetration BC and divergence-free at the same time. Furthermore, to stabilize the method, we have developed a novel density-based particle shifting technique (PST), specifically designed to deal with incompressible fluids. In this formulation, the numerical density is given as one of the most critical constraint variables. As a result, the proposed density-based PST can maintain the fluid's overall volume for the whole simulation. In addition, it also provides numerical stability as it prevents particle clustering and leads the fluid domain to an isotropic composition. First, we verified the proposed corrected formulation with the novel Neumann BC for both non-penetration and non-slip conditions with the simulation of hydrostatic pressure and Poisenuille flow, respectively. Then, we tested the proposed density-based PST with the rotating square patch problem with results comparable to previous studies. Lastly, we verified the proposed method for the dam break with an obstacle test, a highly dynamic problem.

The nonlinear Heisenberg-Euler theory is capable of describing the dynamics of vacuum polarization, a key prediction by quantum electrodynamics. Due to vast progress in the field of laser technology in recent years vacuum polarization can be triggered in the lab by colliding high-intensity laser pulses, leading to a variety of interesting novel phenomena. Since analytical methods for highly nonlinear problems are generally limited and since the experimental requirements for the detection of the signals from the nonlinear quantum vacuum are high, the need for numerical support is apparent. The paper presents a highly-accurate, efficient numerical scheme for solving the nonlinear Heisenberg-Euler equations in weak-field expansion up to six-photon interactions. Properties of the numerical scheme are discussed and an implementation accurate up to order thirteen in terms of spatial resolution is given. Simulations are presented and benchmarked with known analytical results. The versatility of the numerical solver is demonstrated by solving problems in complicated configurations.

This paper presents a computational methodology developed for a high-order approximation of compressible fluid dynamics equations with discontinuities. The methodology is based on a discontinuous Galerkin spectral-element method (DGSEM) built upon a split discretization framework with summation-by-parts (SBP) property, which mimics the integration-by-parts operation in a discrete sense. To extend the split DGSEM framework to discontinuous cases, we implement a shock capturing method based on the entropy viscosity formulation. The developed high-order split-form DGSEM with shock-capturing methodology is subject to a series of evaluation on both one-dimensional and two-dimensional, continuous and discontinuous cases. Convergence of the method is demonstrated both for smooth and shocked cases that have analytical solutions. The 2D Riemann problem tests illustrate an accurate representation of all the relevant flow phenomena, such as shocks, contact discontinuities, and rarefaction waves. All test cases are able to run with a polynomial order of 7 or higher. The values of the tunable parameters related to the entropy viscosity are robust for both 1D and 2D test problems. We also show that higher-order approximations yield smaller errors than lower-order approximations, for the same number of total degrees of freedom.

Scientific machine learning has been successfully applied to inverse problems and PDE discovery in computational physics. One caveat concerning current methods is the need for large amounts of (“clean”) data, in order to characterize the full system response and discover underlying physical models. Bayesian methods may be particularly promising for overcoming these challenges, as they are naturally less sensitive to the negative effects of sparse and noisy data. In this paper, we propose to use Bayesian neural networks (BNN) in order to: 1) Recover the full system states from measurement data (e.g. temperature, velocity field, etc.). We use Hamiltonian Monte-Carlo to sample the posterior distribution of a deep and dense BNN, and show that it is possible to accurately capture physics of varying complexity, without overfitting. 2) Recover the parameters instantiating the underlying partial differential equation (PDE) governing the physical system. Using the trained BNN, as a surrogate of the system response, we generate datasets of derivatives that are potentially comprising the latent PDE governing the observed system and then perform a sequential threshold Bayesian linear regression (STBLR), between the successive derivatives in space and time, to recover the original PDE parameters. We take advantage of the confidence intervals within the BNN outputs, and introduce the spatial derivatives cumulative variance into the STBLR likelihood, to mitigate the influence of highly uncertain derivative data points; thus allowing for more accurate parameter discovery. We demonstrate our approach on a handful of example, in applied physics and non-linear dynamics.

Numerical models for predicting future ice mass loss of the Antarctic and Greenland ice sheets require accurately representing their dynamics. Unfortunately, ice-sheet models suffer from a very strict time-step size constraint, which for higher-order models constitutes a severe bottleneck; in each time step a nonlinear and computationally demanding system of equations has to be solved. In this study, stable time-step sizes are increased for a full-Stokes model by implementing a so-called free-surface stabilization algorithm (FSSA). Previously this stabilization has been used successfully in mantle-convection simulations where a similar viscous-flow problem is solved. By numerical investigation it is demonstrated that instabilities on the very thin domains required for ice-sheet modeling behave differently than on the equal-aspect-ratio domains the stabilization has previously been used on. Despite this, and despite the different material properties of ice, it is shown that it is possible to adapt FSSA to work on idealized ice-sheet domains and increase stable time-step sizes by at least one order of magnitude. The FSSA method presented is deemed accurate, efficient and straightforward to implement into existing ice-sheet solvers.

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions—as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

We present a novel approach to simulating general two-dimensional flows, which could also be applied to other areas of continuum mechanics. The approach generalises the Particle-In-Cell (PIC) method, originally used to model two-dimensional hydrodynamics, by representing fluid elements by elliptical parcels. The rotation and deformation of these parcels are calculated, and parcels split beyond a critical aspect ratio. Conversely, small parcels are eliminated by merging them with larger ones. The elliptical parcels well represent the flow deformation and have excellent conservation properties. In contrast to earlier work that combined PIC with elliptical parcels that split and merge, a vorticity-based framework is used, and accurate integration over ellipses is performed efficiently by two-point Gaussian quadrature. The small-scale mixing associated with parcel splitting and merging is shown to be strongly convergent with grid resolution. The robustness, versatility, accuracy and efficiency of the new Elliptical Parcel-In-Cell (EPIC) method is demonstrated for a variety of standard test cases, and compared with a standard pseudo-spectral method. The results indicate that EPIC is a promising, Lagrangian-based alternative to grid-based methods.

Estimating modeling parameters based on a prescribed optimization target requires to solve an inverse problem, which is commonly ill-posed. Consequently, either infinitely many or no solutions may exist, depending on whether the system is under- or overdetermined, and whether it is consistent or inconsistent. This paper focuses on scenarios where the solution is ambiguous and infinitely many combinations of possible parameter values can accurately achieve the optimization target. Selecting the most suitable solution requires incorporating additional constraints into the model, which is achieved by regularizing the inverse problem. However, common regularization approaches require the specification of a priori unknown regularization hyperparameters that are difficult and tedious to obtain, and can have a large impact on the result.

Here, a novel strategy to reduce the ambiguity of such inverse problems is presented, ensuring that the primary optimization target is always reached accurately. To further reduce the solution space, additional constraints are included, until the optimal modeling parameters are found. Importantly, the required regularization parameters have a direct physical meaning and can be derived sequentially, starting from an initial guess that can be obtained conveniently by solving the system without regularization.

By considering several illustrative examples, the applicability of the method is demonstrated, and its potential for various comparable inverse problems is highlighted.

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much success in developing such methods for finite volume method (FVM) discretizations of PDEs. The Picard Integral formulation (PIF) has recently made such single-stage temporal methods accessible for finite difference method (FDM) discretizations. PIF methods rely on the so-called Lax-Wendroff procedures to tightly couple spatial and temporal derivatives through the governing PDE system to construct high-order Taylor series expansions in time. Going to higher than third order in time requires the calculation of Jacobian-like derivative tensor-vector contractions of an increasingly larger degree, greatly adding to the complexity of such schemes. To that end, we present in this paper a method for calculating these tensor contractions through a recursive application of a discrete Jacobian operator that readily and efficiently computes the needed contractions entirely agnostic of the system of partial differential equations (PDEs) being solved.