Numerical Reference Solutions Research Articles

On Some Boundary Value Methods

Circumventing order restrictions on numerical methods designed for the integration of stiff initial value problem is the concern here via Boundary Value Method. The attainable order p = k+v and linear stability properties of the methods are discussed. The numerical test on some stiff problems shows that the new methods developed, compare favourably with existing methods, with ODE15s of MATLAB used as reference numerical solution.

Impact of Anisotropic Mesh Adaptation on the Aerothermodynamics of Atmospheric Reentry

The presence of complex geometries and/or multiple bodies during atmospheric reentry may lead to complex and directional flow features, such as shock waves and shear layers, that need to be correctly predicted to ensure accurate calculation of the aerothermal loads during reentry. Central in ensuring reliable numerical prediction of loads is the adoption of meshes that ensure grid independence and minimize the misalignment between the directional flow features and the grid cell interfaces, a situation that is known to give rise to nonphysical behaviors and spurious oscillations. The use of anisotropic unstructured grid adaptation is here presented as a means to ensure appropriate grid resolution and alignment with directional flow features for cases where the use of structured grids is not always possible or practical. Results highlight the effectiveness and reliability of anisotropic mesh adaptation in successfully predicting the location of shock discontinuities as well as surface aerothermodynamic quantities, while providing results comparable with established approaches relying on structured meshes. Results are presented for single- and multiple-body cases through comparison with experimental data and reference numerical solutions.

Root System Scale Models Significantly Overestimate Root Water Uptake at Drying Soil Conditions.

Soil hydraulic conductivity (ksoil) drops significantly in dry soils, resulting in steep soil water potential gradients (ψs) near plant roots during water uptake. Coarse soil grid resolutions in root system scale (RSS) models of root water uptake (RWU) generally do not spatially resolve this gradient in drying soils which can lead to a large overestimation of RWU. To quantify this, we consider a benchmark scenario of RWU from drying soil for which a numerical reference solution is available. We analyze this problem using a finite volume scheme and investigate the impact of grid size on the RSS model results. At dry conditions, the cumulative RWU was overestimated by up to 300% for the coarsest soil grid of 4.0 cm and by 30% for the finest soil grid of 0.2 cm, while the computational demand increased from 19 s to 21 h. As an accurate and computationally efficient alternative to the RSS model, we implemented a continuum multi-scale model where we keep a coarse grid resolution for the bulk soil, but in addition, we solve a 1-dimensional radially symmetric soil model at rhizosphere scale around individual root segments. The models at the two scales are coupled in a mass-conservative way. The multi-scale model compares best to the reference solution (−20%) at much lower computational costs of 4 min. Our results demonstrate the need to shift to improved RWU models when simulating dry soil conditions and highlight that results for dry conditions obtained with RSS models of RWU should be interpreted with caution.

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

We present a novel staggered semi-implicit hybrid finite volume / finite element method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh composed of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type finite volume method on the staggered dual grid. Next, a classical continuous finite element scheme provides the free surface elevation at the vertices of the primal simplex mesh. The semi-implicit strategy followed in this paper circumvents the contribution of the surface wave celerity to the CFL-type time step restriction, hence making the proposed algorithm well-suited for the solution of low Froude number flows. It can be shown that in the low Froude number limit the proposed algorithm reduces to a semi-implicit hybrid FV/FE projection method for the incompressible Navier-Stokes equations. At the same time, the conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme can be considered an all-Froude number solver, able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and is also validated via numerical experiments. The numerical solution of several Riemann problems, including flat bottom and a bottom with jumps, attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions obtained for the 3D free surface Navier-Stokes equations and with available experimental reference data.

Heat transport in entropy-optimized flow of viscoelastic fluid due to Riga plate: analysis of artificial neural network

In this paper, we focus to design intelligence numerical computing through artificial neural networks (ANNs) which are backpropagated with Levenberg–Marquard technique (NN-BLMT) for the theoretical approach of physical aspects of heat generation in second-grade fluid (PA-HG-SGF) due to Riga plate. The Riga plate (RP) is known as the physical interaction stimulus, which consists of electrodes and enduring electromagnets that are located on plane surface. The purpose of NN-BLMT is to learn the weight of neural networks on the basis of optimization of the fitness value based on mean-square error between the proposed result and reference numerical solution. The innovation and reliability of NN-BLMT used will be greatly better as compared to traditional numerical techniques that are used to solve commercial and industrial problems. NN-BLMT is fast and easy to apply on nonlinear problems and get best results. The original model PA-HG-SGF in term of PDEs is first converted into system of nonlinear ODEs through suitable transformation and then numerically solved. Through Adam numerical technique (AMT) in Mathematica software, a dataset for PA-HG-SGF is attained for different scenarios of PA-HG-SGF by variation of second-grade parameter, heat-generation parameter, Hartman number, thermal lamination parameter, thermal composure parameter and Prandtl number. Expected solutions are described for PA-HG-SGF through the NN-BLMT testing, training, and validation process. In addition, NN-BLMT relative studies and performance analysis are validated by histogram studies, regression analysis and MSE and then analyzed PA-HG-SGF.

A nonlinear piezoelectric shunt absorber with a 2:1 internal resonance: Theory

In this paper, a semi-passive nonlinear piezoelectric shunt absorber is presented, aiming at attenuating the vibration of a resonant elastic structure under external excitation. This is done by connecting the elastic structure to a nonlinear shunt circuit via a piezoelectric patch. The nonlinear shunt circuit consists of resonant circuit that includes a quadratic non-linearity. A particular tuning of the natural frequency of the shunt enables to create a two to one internal resonance. This generates a strong coupling between the mechanical mode and the electrical mode, leading to replace the mechanical resonance with a nonlinear antiresonance associated with an amplitude saturation, thus leading to an efficient vibration reduction. In this paper, we first propose a theoretical model that is expanded onto a suitable electromechanical modal basis and reduced to the two modes of interest, nonlinearly coupled by quadratic terms. Then, analytical solutions are obtained by the multiple scale method and compared to a reference numerical solutions stemming from the harmonic balance method. This enables to investigate the performance of the system in term of vibration absorption as well as giving design rules to tune the nonlinear shunt and to choose the free parameters of the system.

Potentials of the Application of the Electrical Transmission Line Theory for Thermal Investigations on Cables

Abstract. In this contribution, similarities and differences between electrical and thermal effects on cables are investigated. In the electrical transmission line theory, a wide variety of methods is known to describe the voltage and current along cables. The potential for the adaption of some of those methods to thermal problems is discussed. Exemplarily, for an unshielded single cable, an analytical solution based on the Laplace transform and an approach based on cascaded equivalent circuits are compared with a numerical reference solution and measurement results.

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

Simulation of borehole acoustic wavefields in fractured media by combining the spectral-element method and linear-slip model

We have used the spectral-element method (SEM) to simulate 3D acoustic wavefields in the fluid-filled borehole embedded in fractured media. The fractures are characterized by the linear-slip model (LSM), which is incorporated into the surface integral of the SEM weak form, avoiding meshing individual fractures, thus reducing the degrees of freedom of the fractures compared with meshing each fracture directly. For the fracture-free case, we validate SEM through the comparison with the real-axis integration method for monopole and dipole sources. For the case with a fracture, we compare the SEM-LSM solutions with the reference numerical solutions of a thin-layer model using the finite-difference method. Good agreement is achieved between the results from our method and the reference finite-difference solutions. We find that the acoustic wavefields excited by a dipole source are more sensitive to the fractures than those by a monopole source. To show the ability of the approach to handle complex problems, we simulate the cases with a tilted fracture and multiple fractures. Based on the simulated results, we investigate the influence of the fracture parameters (e.g., stiffness, tilt angle, azimuth, thickness, number, and spatial intervals of fractures) on the scattered wavefields. We find that the tilt angle has an obvious influence on the scattered waveforms and amplitudes. The results also demonstrate that the wavefields are quite sensitive to the number of fractures. The magnitudes of the horizontal components’ transmitted wavefields decrease linearly with the number of fractures. Through analyzing the synthetic data in the time and frequency domains, we discuss how to evaluate the properties of fractures intersected by a borehole.

Comparison of Different Numerical Interface Capturing Methods for the Simulation of Faraday Waves

Faraday instability is a classic problem that occurs due to the relative displacement of the interface that separates two immiscible fluids placed in a closed container under oscillating acceleration parallel to gravity. The interface deformation and the induced flow patterns of this two-phase flow are very complex and numerical simulations could allow a deeper understanding of the dynamics of these systems. Some tests have been performed to establish a reference solution, but further validation is needed in order to ensure the validity of these solutions. In this work, we compare some numerical solutions for the linear and nonlinear regimes using the phase field scheme with predictions obtained using different numerical schemes such as Front Tracking, Volume of Fluid, and Element-based Finite Volume Method. The results show that, in both linear and nonlinear regimes, some important differences in the prediction of the interface dynamics between the methods are observed, and the need to provide a reference numerical solution for future benchmarks is highlighted.

A Numerical Solution for Modelling Mooring Dynamics, Including Bending and Shearing Effects, Using a Geometrically Exact Beam Model

During the operation of moored, floating devices in the renewable energy sector, the tight coupling between the mooring system and floater motion results in snap load conditions. Before snap events occur, the mooring line is typically slack. Here, the mechanism of energy propagation changes from axial to bending dominant, and the correct modelling of the rotational deformation of the lines becomes important. In this paper, a new numerical solution for modelling the mooring dynamics that includes bending and shearing effects is proposed for this purpose. The approach is based on a geometrically exact beam model and quaternion representations for the rotational deformations. Further, the model is coupled to a two-phase numerical wave tank to simulate the motion of a moored, floating offshore wind platform in waves. A good agreement between the proposed numerical model and reference solutions was found. The influence of the bending stiffness on the motion of the structure was studied subsequently. We found that increased stiffness increased the amplitudes of the heave and surge motion, whereas the motion frequencies were less altered.

An Analytical Solution for Pressure-Induced Deformation of Anisotropic Multilayered Subsurface

We present a generalized Geertsma solution that can consider any number of finite-thickness layers in the subsurface whose mechanical properties are different from layer to layer. In addition, each layer can be assumed either isotropic or anisotropic. The accuracy of the generalized solution is validated against a numerical reference solution. The generalized Geertsma solution is further extended by a linear superposition framework that enables a response simulation due to an arbitrarily-distributed non-uniform pressure anomaly. The linear superposition approach is tested and validated by solving a realistic synthetic model based on the In Salah CO2 storage model and compared with a full 3D finite element solution. Finally, by means of a simple inversion exercise (based on the linear superposition approach), we learn that the stiffnesses of cap rock and reservoir are the most influencing parameter on the inversion result for a given layering geometry, suggesting that it is very important to estimate high-confidence mechanical properties of both cap rock and reservoir.

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics

We present the application of a class of deep learning, known as Physics Informed Neural Networks (PINN), to inversion and surrogate modeling in solid mechanics. We explain how to incorporate the momentum balance and constitutive relations into PINN, and explore in detail the application to linear elasticity, and illustrate its extension to nonlinear problems through an example that showcases von Mises elastoplasticity. While common PINN algorithms are based on training one deep neural network (DNN), we propose a multi-network model that results in more accurate representation of the field variables. To validate the model, we test the framework on synthetic data generated from analytical and numerical reference solutions. We study convergence of the PINN model, and show that Isogeometric Analysis (IGA) results in superior accuracy and convergence characteristics compared with classic low-order Finite Element Method (FEM). We also show the applicability of the framework for transfer learning, and find vastly accelerated convergence during network re-training. Finally, we find that honoring the physics leads to improved robustness: when trained only on a few parameters, we find that the PINN model can accurately predict the solution for a wide range of parameters new to the network—thus pointing to an important application of this framework to sensitivity analysis and surrogate modeling.

Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks

Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

Nonlinear deterministic sea wave prediction using instantaneous velocity profiles

Optimizing the production of wave energy converters using Model Predictive Control (MPC) requires a real-time, deterministic prediction of the waves arriving at the device. This study presents a new method for deterministic sea wave prediction, using the horizontal velocity profile over the water column as a boundary condition for a dedicated nonlinear wave model. However, direct measurement of the horizontal velocity component over the whole vertical column is hardly achievable at sea. A method to reconstruct this profile from measurement devices currently at use, such as ADCPs, is thus presented and evaluated. The performance of the prediction method itself is then tested using synthetic numerical data. First, the reconstruction of the horizontal velocity profile as a boundary condition is evaluated. Then, the whole prediction procedure is assessed. In both these stages, the simulations are based on synthetic numerical data and the outcomes are compared with numerical reference solutions. The results show that the method is promising enough to justify further investigation through wave tank experiments.

Discontinuous Galerkin solution of the RANS and [formula omitted] equations for natural and bypass transition

Transition modelling represents a key ingredient for improving the performance predictions of many industrial applications. Among transition models, local formulations seem to guarantee better robustness, accuracy and easiness of implementation in modern CFD solvers. These models have been proposed in the finite volume context to predict the laminar-turbulent transition, but only few attempts have been made in the high-order framework. In this paper a new phenomenological transition model based on the concept of laminar kinetic energy has been proposed and implemented in a high-order accurate Discontinuous Galerkin code, named MIGALE. The transition model is validated and assessed by computing the transitional flow around flat plates with zero/adverse pressure gradients and through different turbine nozzles (T106A and LS89) for different values of Reynolds number and turbulence intensity. The computed results have been compared with experimental data and reference numerical solutions.

A Hybrid-Dimensional Coupled Pore-Network/Free-Flow Model Including Pore-Scale Slip and Its Application to a Micromodel Experiment

Modeling coupled systems of free flow adjacent to a porous medium by means of fully resolved Navier–Stokes equations is limited by the immense computational cost and is thus only feasible for relatively small domains. Coupled, hybrid-dimensional models can be much more efficient by simplifying the porous domain, e.g., in terms of a pore-network model. In this work, we present a coupled pore-network/free-flow model taking into account pore-scale slip at the local interfaces between free flow and the pores. We consider two-dimensional and three-dimensional setups and show that our proposed slip condition can significantly increase the coupled model’s accuracy: compared to fully resolved equidimensional numerical reference solutions, the normalized errors for velocity are reduced by a factor of more than five, depending on the flow configuration. A pore-scale slip parameter beta _{{{{rm pore}}}} required by the slip condition was determined numerically in a preprocessing step. We found a linear scaling behavior of beta _{{{{rm pore}}}} with the size of the interface pore body for three-dimensional and two-dimensional domains. The slip condition can thus be applied without incurring any run-time cost. In the last section of this work, we used the coupled model to recalculate a microfluidic experiment where we additionally exploited the flat structure of the micromodel which permits the use of a quasi-3D free-flow model. The extended coupled model is accurate and efficient.

An ultra-fast single phase reservoir model for multi-well developments

This paper addresses the problem of reservoir modeling in the context of large scale developments, in which multiple wells with hundreds of fractures are placed in a formation. In particular, the goal is to develop an ability to compute the production rate for each well for the given bottomhole pressure variation and also to evaluate the associated change of stress in the reservoir, e.g. for the purpose of subsequent modeling of hydraulic fracturing in neighboring wells. In addition, it is desired to compute the solution rapidly, in which case this makes it possible to consider sensitivities of the solution to various parameters. In order to achieve the goal, i.e. having an ability to solve large scale problems quickly, it is necessary to consider a series of simplifications in the model. The primary assumptions are zero vertical permeability and single phase flow, which makes the flow equations linear and decoupled for each layer. In addition, the use of an analytical solution that captures linear flow for imposing the boundary condition at the fracture surface allows to avoid having fine discretization near the fracture surface. Finally, the use of a uniform grid together with employing Fast Fourier Transform algorithm and Graphics Processing Unit for calculations leads to the desired performance. The model has been benchmarked by comparing its predictions to reference numerical and analytical solutions. Computational performance of the algorithm is analyzed and it is shown that it is possible to simulate large scale problems within minutes. Finally, a field scale example with eight wells is presented, which illustrates a potential application of the algorithm and quantifies the characteristic variations of pressure and stress caused by injection of the fracturing fluid and production.