Nonlinear Time-dependent Problems Research Articles

Multiharmonic multiscale modelling in 3-D nonlinear magnetoquasistatics: Composite material made of insulated particles

The use of the classical finite element method (FEM) to solve problems with magnetic composites leads to huge linear systems that are impossible to solve. Instead, homogenization and multiscale methods are often used with the composite material replaced by a homogeneous material with the homogenized constitutive law obtained by solving cell-problems representing the mesoscale material structure. For non-linear time-dependent problems, FEM is often used with a time-transient method (TTM) and the solution is obtained one time-step at a time. However, in cases where a steady-state solution is of interest, the multiharmonic method can be faster and more cost effective for the same accuracy of the time discretization. In addition, when solving magnetoquasistatic multiscale problems with TTM, the dynamic hysteresis in the homogenized fields can slow down or even impede the convergence of the macro-scale problem due to the possibly non-continuously differentiable homogenized material laws. This work presents a novel robust modelling approach for non-linear magnetoquasistatic problems combining multiharmonic method with the multiscale method.

Non-intrusive data-driven reduced-order modeling for time-dependent parametrized problems

Reduced-order models are indispensable for multi-query or real-time problems. However, there are still many challenges to constructing efficient ROMs for time-dependent parametrized problems. Using a linear reduced space is inefficient for time-dependent nonlinear problems, especially for transport-dominated problems. The non-linearity usually needs to be addressed by hyper-reduction techniques, such as DEIM, but it is intrusive and relies on the assumption of affine dependence of parameters. This paper proposes and studies a non-intrusive reduced-order modeling approach for time-dependent parametrized problems. It is purely data-driven and naturally split into offline and online stages. During the offline stage, a convolutional autoencoder, consisting of an encoder and a decoder, is trained to perform dimensionality reduction. The encoder compresses the full-order solution snapshots to a nonlinear manifold or a low-dimensional reduced/latent space. The decoder allows the recovery of the full-order solution from the latent space. To deal with the time-dependent problems, a high-order dynamic mode decomposition (HODMD) is utilized to model the trajectories in the latent space for each parameter. During the online stage, the HODMD models are first utilized to obtain the latent variables at a new time, then interpolation techniques are adopted to recover the latent variables at a new parameter value, and the full-order solution is recovered by the decoder. Some numerical tests are conducted to show that the approach can be used to predict the unseen full-order solution at new times and parameter values fast and accurately, including transport-dominated problems.

Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores: a strain gradient plate model approach

The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having lattice-type metamaterial cores. The proposed model provides a modelling framework computationally more efficient than framework of the conventional three-dimensional elasticity for the simulation of the underlying nonlinear dynamics of cellular plates with microarchitectures. The corresponding governing equations follow Mindlin’s strain gradient elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order C1-continuous 6-node finite element is adopted for the discretization in the spatial domain, and an arc-length continuation technique along with time-periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as reliable validation references for triangularly prismatic example cores, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response. The results show that the proposed strain gradient plate model can accurately capture the microarchitecture effects of cellular plates with low computational costs, which signifies a relevant contribution particularly to the computational analysis of nonlinear dynamics.

An Extrinsic Approach Based on Physics-Informed Neural Networks for PDEs on Surfaces

In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the extrinsic approach with PINNs and then express that loss function in extrinsic form. Subsequently, the loss function could be minimized extrinsically with respect to the NN parameters. Numerical results demonstrate that the extrinsic approach based on PINNs for surface problems has good accuracy and higher efficiency compared with the embedding approach based on PINNs. In addition, the strong nonlinear mapping ability of NNs makes this approach robust in solving time-dependent nonlinear problems on more complex surfaces.

Reduced basis methods for time-dependent problems

Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. However, many challenges remain to secure the robustness and efficiency needed for the numerical simulation of nonlinear time-dependent problems. The purpose of this article is to survey the state of the art of reduced basis methods for time-dependent problems and draw together recent advances in three main directions. First, we discuss structure-preserving reduced order models designed to retain key physical properties of the continuous problem. Second, we survey localized and adaptive methods based on nonlinear approximations of the solution space. Finally, we consider data-driven techniques based on non-intrusive reduced order models in which an approximation of the map between parameter space and coefficients of the reduced basis is learned. Within each class of methods, we describe different approaches and provide a comparative discussion that lends insights to advantages, disadvantages and potential open questions.

POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations

AbstractIn the present paper we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable, e.g., in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

Projection-based reduced order models for parameterized nonlinear time-dependent problems arising in cardiac mechanics

<abstract><p>The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.</p></abstract>

Adaptive preconditioning for a stream of SLAEs

The paper considers the way of reducing the time consumed to solve SLAEs with iterative methods by reusing the data structures obtained in the solution of a previous SLAE, or selecting a preconditioner from the available set of preconditioners to minimize the time of solving the next SLAEs. Such adaptive preconditioning is used to solve time-dependent nonlinear problems. SLAEs generated at the Newton iteration n-1 of every computation step are solved using the SLAE structure of the first Newton iteration and the selection of a preconditioner from the given set allows reducing the time of solving SLAEs of a varying complexity at different time steps. The adaptive preconditioning idea and its application are demonstrated for a stream of SLAEs in some RFNC-VNIIEF’s codes.

Numerical analysis of nonlinear functionally graded piezoelectric energy harvester subjected to multi moving oscillators

In this paper, energy harvesting from nonlinear vibration of a functionally graded beam covered by a piezoelectric patch under multi-moving oscillators is studied. The material of both the substructure and the piezoelectric patch is assumed to be functionally graded in the thickness direction. A coupled system of equations considering Euler-Bernoulli beam theory and von-Karman nonlinearity as well as electromechanical coupling are derived using the generalized Hamilton’s principle. Finite element method as well as Newmark time integration scheme are used to solve the coupled nonlinear time dependent problem. The effects of different parameters including material distribution, velocity of the moving oscillators, piezoelectric patch thickness and load resistance on the output voltage and harvested power are investigated. Moreover, the effects of oscillator characteristics such as damping ratio and stiffness on the nonlinear behavior of the beam and harvested power are also studied. Results indicate that the aforementioned parameters have considerable effects on the harvested power. It is also shown that ignoring nonlinear effects may lead to erroneous and unacceptable results. To the best of authors’ knowledge, there is no study about energy harvesting from nonlinear vibration of beams under moving oscillators.

A mixed formulation of proper generalized decomposition for solving the Allen-Cahn and Cahn-Hilliard equations

In this paper, we propose a new implementation strategy for solving the Allen-Cahn and Cahn-Hilliard equations with the proper generalized decomposition (PGD) method for parametric studies. As is common to all PGD methods, it is not necessary to do essentially repeating computations; instead, solutions for a range of parameters can be obtained in a single computation. The proposed implementation strategy includes a mixed formulation and a new data structure. The mixed formulation is applied with the staggered method which does not require the first variation of residual with the derivative of the bulk free energy and thereby avoid repeated matrix reassembling in the Newton-Raphson procedure. In addition, the proposed data structure stores pointwise values of the solution functions instead of the PGD representation for the residual of the Newton method, which is efficient for nonlinear time-dependent problems. Overall, the computational efficiency is significantly improved compared with most traditional PGD formulations and implementations.

A multi-stage deep learning based algorithm for multiscale model reduction

In this work, we propose a multi-stage training strategy for the development of deep learning algorithms applied to problems with multiscale features. Each stage of the proposed strategy shares an (almost) identical network structure and predicts the same reduced order model of the multiscale problem. The output of the previous stage will be combined with an intermediate layer for the current stage. We numerically show that using different reduced order models as inputs of each stage can improve the training and we propose several ways of adding different information into the systems. These methods include mathematical multiscale model reductions and network approaches; but we found that the mathematical approach is a systematical way of decoupling information and gives the best result. We finally verified our training methodology on a time dependent nonlinear problem and a steady state model.

Numerical Solutions of Time Dependent Partial Differential Equations via HAAR Wavelets

An effective wavelet based scheme coupled with finite difference is used for the solution of two nonlinear time dependent problems namely: Burgers' and Boussinesq equations. These equations have wide-spread application in many fields such as viscous medium, turbulence , uid dynamics, infiltration phenomena etc. The proposed scheme convert the partial differential equations (PDE) to system of algebraic equations. The obtained system can be solved easily. In this paper convergence of the scheme is also discussed to show validity of the technique. Effectiveness of the scheme is shown with the help of test problems. Numerical results verify that the suggested scheme is more accurate, convenient, fast and require low computational cost.

A Study on the LATIN-PGD Method: Analysis of Some Variants in the Light of the Latest Developments

The LATIN-PGD method is a powerful alternative to the Newton–Raphson scheme for solving non-linear time-dependent problems in combination with reduced-order modeling methods. Many developments have been carried out over the last few decades and have led to some variants of the LATIN-PGD method. However, only few comparisons have been made between these variants and none using the digital resources now available. In this article, a comparison between the two major variants of the LATIN-PGD method, as well as with the Newton–Raphson one, is performed using a unified software which highlights the assets of the LATIN-PGD. Various test cases dealing with elasto-visco-plastic problems are undertaken, including comparison with commercial solvers, which reveals the interesting time saving in favor of the LATIN-PGD method.

A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO

We provide a new approach for the efficient matrix-free application of the transpose of the Jacobian for the spectral element method for the adjoint-based solution of partial differential equation (PDE) constrained optimization. This results in optimizations of nonlinear PDEs using explicit integrators where the integration of the adjoint problem is not more expensive than the forward simulation. Solving PDE constrained optimization problems entails combining expertise from multiple areas, including simulation, computation of derivatives, and optimization. The Portable, Extensible Toolkit for Scientific computation (PETSc) together with its companion package, the Toolkit for Advanced Optimization (TAO), is an integrated numerical software library that contains an algorithmic/software stack for solving linear systems, nonlinear systems, ordinary differential equations, differential algebraic equations, and large-scale optimization problems and, as such, is an ideal tool for performing PDE-constrained optimization. This paper describes an efficient approach in which the software stack provided by PETSc/TAO can be used for large-scale nonlinear time-dependent problems. Time integration can involve a range of high-order methods, both implicit and explicit. The PDE-constrained optimization algorithm used is gradient-based and seamlessly integrated with the simulation of the physical problem.

Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability

The pollution of groundwater, essential for supporting populations and agriculture, can have catastrophic consequences. Thus, accurate modeling of water pollution at the surface and in groundwater aquifers is vital. Here, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. Addressing this problem is relevant for geothermal reservoir simulations, natural saline-disposal basins, modeling of contaminant plumes and subsurface flow predictions. This strongly nonlinear time-dependent problem describes the convection of a two-phase flow, whereby a liquid flows and propagates into groundwater reservoirs under the force of gravity to form so-called “fingers”. To achieve an accurate numerical solution, fine spatial resolution with an unstructured mesh and, therefore, high computational resources are required. Here we run a parallelized simulation toolbox ug4 with a geometric multigrid solver on a parallel cluster, and the parallelization is carried out in physical and stochastic spaces. Additionally, we demonstrate how the ug4 toolbox can be run in a black-box fashion for testing different scenarios in the density-driven flow. As a benchmark, we solve the Elder-like problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities for the mass fraction. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

Partition of Unity Finite Element Analysis of Nonlinear Transient Diffusion Problems Using p-Version Refinement

We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems. The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods. A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated. However, these enrichment functions have been used only in context of the standard h-version refinement or the so-called q-version refinement. In this paper we demonstrate that the p-version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method. First, the transient diffusion problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the p-version enriched finite elements. Numerical results are presented for three test examples of timedependent diffusion problems in both homogeneous and heterogeneous media. The computed results show the significant improvement when using the p-version refined enriched approximations in the finite element analysis. In addition, these results support our expectations for a robust and high-order accurate enriched partition of unity finite element method.

Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls

• We consider the temporal stability of multiple solutions for flows in a porous channel with moving walls for the first time. • We construct second order finite difference schemes for eigenproblems associated with the linear stability analysis . • We verify the linear stability results for the first time by solving the perturbed nonlinear time dependent equation. In this paper, a viscous, incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is considered. We present multiple symmetric steady-state solutions of this flow problem at several different expanding ratios, and use the linear stability theory to analyse the temporal stability for these solutions under symmetric, antisymmetric and general perturbations. We construct second order finite difference schemes for the eigenvalue problems with boundary conditions associated with those perturbations, and observe that most of these solutions which are stable under symmetric perturbations are unstable under antisymmetric perturbations. Furthermore, we verify the linear stability analysis results by directly solving the original perturbed nonlinear time dependent problem, and find that both stability results are consistent.

Topological sensitivity analysis of a time-dependent nonlinear problem

We are interested in the optimization of the pipe shape allowing minimization of the dissipated energy in time-dependent Navier–Stokes Darcy flow. The used technique is based on the topological gradient method. In the theoretical part, we present an analysis of the topological sensitivity for the dissipated energy function. Some numerical tests are presented to illustrate the developed approach.

Superconvergence analysis of nonconforming FEM for nonlinear time-dependent thermistor problem

In this paper, we study the superconvergence analysis of nonlinear time-dependent thermistor problem with the well-known nonconforming element, i.e., the extension of the rotated bilinear element (denoted EQ1rot), for the semi-discrete and a linearized backward Euler fully-discrete schemes. The superclose and superconvergent estimates about the related variables in broken H1-norm are derived with the help of the rigorous analysis together with the special properties of this element, mean value technique and interpolated post-processing approach. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

An accurate meshless formulation for the simulation of linear and fully nonlinear advection diffusion reaction problems

In this paper, a new meshless semi-analytical collocation technique is presented for the simulation of time-dependent linear and fully nonlinear advection diffusion reaction equations (ADREs). The time-dependent nonlinear problem is reduced to the system of linear problems in each time layer by using the quasilinearization technique in combination with the Crank–Nicolson scheme for the temporal approximation. Then it may be solved by the meshless backward substitution method. In the backward substitution method the approximation is given to satisfy the boundary conditions as a prior. Then the approximation is substituted back to the governing equations where the unknown weighted parameters are obtained using the collocation approach. The developed method has been tested on seven concrete problems for a range of free parameters. The comparison of the obtained numerical results with those available in literatures reveals that the proposed method can accurately approximate linear and nonlinear advection diffusion reaction problems. This method can also be easily extended onto other linear and nonlinear transient problems in 1D, 2D, and 3D due to its general idea and easy implementation.