arXiv subjects

Eirik Valseth

Publications and source records attributed to Eirik Valseth.

30 records · Page 2Linked to original sources

arXiv2023

WAVEx: Stabilized Finite Elements for Spectral Wind Wave Models Using FEniCSx

The prediction of the wind wave spectrum of the ocean using numerical models are an important tool for researchers, engineers, and communities living in coastal areas. The governing equation of the wind wave models, the Wave Action Balance Equation, presents unique challenges for implementing reliable numerical models because it is highly advective, highly nonlinear and high dimensional. Historically, most operational models have utilized finite difference methods, others have used finite volume methods but relatively few attempts at using finite element methods. In this work, we seek to fill this gap by investigating several different finite element discretizations of the Wave Action Balance Equation. The methods, which include streamline upwind Petrov-Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented and convergence properties are examined for some simplified 2-D test cases. Then, a new spectral wind wave model, WAVEx, is formulated and implemented for the full problem setting. WAVEx uses continuous finite elements along with SUPG stabilization in geographic/spectral space that allows for fully unstructured triangular meshes in both geographic and spectral space. For propagation in time, a second order fully implicit finite difference method is used. When source terms are active, a second order operator splitting scheme is used to linearize the problem. In the splitting scheme, propagation is solved using the implicit method and the nonlinear source terms are treated explicitly. Several test cases, including analytic tests and laboratory experiments, are demonstrated and results are compared to analytic solutions, observations, as well as output from another model that is used operationally.

physics.flu-dyn↗

arXiv2023

Application of a dispersive wave hydro-sediment-morphodynamic model in the Ria Formosa lagoon

Results of an application of a dispersive wave hydro-sediment-morphodynamic model in the western circulation cell of the Ria Formosa lagoon located in the Algarve region of the southern Portugal are presented. This area of interest has a couple of features that complicate the application of the dispersive wave model: (1) the area has a complex irregular geometry with a number of barrier islands that separate the lagoon from the Atlantic Ocean, artificial and naturally occurring tidal inlets, and a number of curling channels inside the lagoon that interconnect the inlets and serve as waterways between the lagoon settlements; (2) the tidal range in the area can reach up to 3.5 m; therefore, the terrain inside the lagoon is characterized by vast salt marshes and tidal flats, and the wetting-drying process is a key component of any hydrodynamic simulation in this area. A model representation of the area has been developed by generating an unstructured finite element mesh of the circulation cell, and collecting data on parameters that characterize the tidal waves in the area, and bottom friction and sediment transport models used in the simulations. The results of the simulations indicate that the dispersive wave model can be applied in coastal areas with nontrivial underlying physical processes, and complex irregular geometries. Moreover, the dispersive term of the model is capable of capturing additional flow characteristics that are otherwise not present in hydrodynamic simulations that involve the nonlinear shallow water equations; and these additional flow features can, in their turn, affect the resulting sediment transport and bed morphodynamic process simulations.

physics.geo-ph↗

arXiv2023

A Framework for Flexible Peak Storm Surge Prediction

Storm surge is a major natural hazard in coastal regions, responsible both for significant property damage and loss of life. Accurate, efficient models of storm surge are needed both to assess long-term risk and to guide emergency management decisions. While high-fidelity regional- and global-ocean circulation models such as the ADvanced CIRCulation (ADCIRC) model can accurately predict storm surge, they are very computationally expensive. Here we develop a novel surrogate model for peak storm surge prediction based on a multi-stage approach. In the first stage, points are classified as inundated or not. In the second, the level of inundation is predicted . Additionally, we propose a new formulation of the surrogate problem in which storm surge is predicted independently for each point. This allows for predictions to be made directly for locations not present in the training data, and significantly reduces the number of model parameters. We demonstrate our modeling framework on two study areas: the Texas coast and the northern portion of the Alaskan coast. For Texas, the model is trained with a database of 446 synthetic hurricanes. The model is able to accurately match ADCIRC predictions on a test set of synthetic storms. We further present a test of the model on Hurricanes Ike (2008) and Harvey (2017). For Alaska, the model is trained on a dataset of 109 historical surge events. We test the surrogate model on actual surge events including the recent Typhoon Merbok (2022) that take place after the events in the training data. For both datasets, the surrogate model achieves similar performance to ADCIRC on real events when compared to observational data. In both cases, the surrogate models are many orders of magnitude faster than ADCIRC.

cs.CE↗

arXiv2023

Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems

We establish stable finite element (FE) approximations of convection-diffusion initial boundary value problems using the automatic variationally stable finite element (AVS-FE) method. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered singular perturbation in both space and time. The unconditional stability of the AVS-FE method, regardless of the underlying differential operator, allows us significant flexibility in the construction of FE approximations. We take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-alpha method. In the generalized-alpha method, we discretize the temporal domain into finite sized time-steps and adopt the generalized-alpha method as time integrator. Then, we derive a corresponding norm for the obtained operator to guarantee the temporal stability of the method. We present numerical verifications for both approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin method by Demkowicz and Gopalakrishnan, the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serve as error indicators in both space and time for which we present multiple numerical verifications adaptive strategies.

math.NA↗

arXiv2022

Cross-mode Stabilized Stochastic Shallow Water Systems Using Stochastic Finite Element Methods

The development of surrogate models to study uncertainties in hydrologic systems requires significant effort in the development of sampling strategies and forward model simulations. Furthermore, in applications where prediction time is critical, such as prediction of hurricane storm surge, the predictions of system response and uncertainties can be required within short time frames. Here, we develop an efficient stochastic shallow water model to address these issues. To discretize the physical and probability spaces we use a Stochastic Galerkin method and a Incremental Pressure Correction scheme to advance the solution in time. To overcome discrete stability issues, we propose cross-mode stabilization methods which employs existing stabilization methods in the probability space by adding stabilization terms to every stochastic mode in a modes-coupled way. We extensively verify the developed method for both idealized shallow water test cases and hindcasting of past hurricanes. We subsequently use the developed and verified method to perform a comprehensive statistical analysis of the established shallow water surrogate models. Finally, we propose a predictor for hurricane storm surge under uncertain wind drag coefficients and demonstrate its effectivity for Hurricanes Ike and Harvey.

physics.comp-ph↗

arXiv2022

Extending FEniCS to Work in Higher Dimensions Using Tensor Product Finite Elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space-time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection-diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov-Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term. In all cases, a p=1 basis is used and optimal L^2 convergence rates of order h^{p+1} of the errors are achieved with respect to h refinement

math.NA↗

arXiv2021

A Stable Mixed FE Method for Nearly Incompressible Linear Elastostatics

We present a new, stable, mixed finite element (FE) method for linear elastostatics of nearly incompressible solids. The method is the automatic variationally stable FE (AVS-FE) method of Calo, Romkes and Valseth, in which we consider a Petrov-Galerkin weak formulation where the stress and displacement variables are in the space H(div)xH1, respectively. This allows us to employ a fully conforming FE discretization for any elastic solid using classical FE subspaces of H(div) and H1. Hence, the resulting FE approximation yields both continuous stresses and displacements. To ensure stability of the method, we employ the philosophy of the discontinuous Petrov-Galerkin (DPG) method of Demkowicz and Gopalakrishnan and use optimal test spaces. Thus, the resulting FE discretization is stable even as the Poisson ratio approaches 0.5, and the system of linear algebraic equations is symmetric and positive definite. Our method also comes with a built-in a posteriori error estimator as well as well as indicators which are used to drive mesh adaptive refinements. We present several numerical verifications of our method including comparisons to existing FE technologies.

math.NA↗

arXiv2021

A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation

In its application to the modeling of a mineral separation process, we propose the numerical analysis of the Cahn-Hilliard equation by employing spacetime discretizations of the automatic variationally stable finite element (AVS-FE) method. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan. The trial space, however, consists of globally continuous Hilbert spaces such as H1 and H(div). Hence, the AVS-FE approximations employ classical C0 or Raviart-Thomas FE basis functions. The optimal test functions guarantee the numerical stability of the AVS-FE method and lead to discrete systems that are symmetric and positive definite. Hence, the AVS-FE method can solve the Cahn-Hilliard equation in both space and time without a restrictive CFL condition to dictate the space-time element size. We present numerical verifications of both one and two dimensional problems in space. The verifications show optimal rates of convergence in L2 and H1 norms. Results for mesh adaptive refinements using a built-in error estimator of the AVS-FE method are also presented.

math.NA↗

arXiv2020

An Adaptive Stable Space-Time FE Method for the Shallow Water Equations

We consider the finite element (FE) approximation of the shallow water equations (SWE) by considering discretizations in which both space and time are established using an unconditionally stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to break the test space and achieve unconditionally stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We derive a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptivity on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to commonly applied benchmarks for the SWE.

math.NA↗

arXiv2020

Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems

We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS FE) method for scalar-valued convection-diffusion problems. The AVS-FE method is a Petrov-Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov Galerkin (DPG) method by Demkowicz and Gopalakrishnan, leading to unconditionally stable FE approximations. Remarkably, by using C0 or Raviart Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element by element fashion. To establish the error estimators we present two approaches: i) following the classical approach of Becker and Rannacher, i.e., the dual solution is sought in the (broken) test space, and ii) introducing an alternative approach in which we seek C0, or Raviart Thomas, AVS FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for h adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero.

math.NA↗

arXiv2020

An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation

We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg-de Vries (KdV) equation. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov- Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial boundary value problem (IBVP) and a Riesz representer of the approximation error. The AVS-FE method distinguishes itself from other minimum residual methods by using globally continuous Hilbert spaces, such as H1, while at the same time using broken Hilbert spaces for the test. Consequently, the AVS-FE approximations are classical C0 continuous FE solutions. The unconditional stability of this method allows us to solve the KdV equation space and time without having to satisfy a CFL condition. We present several numerical verifications for both linear and nonlinear versions of the KdV equation leading to optimal convergence behavior. Finally, we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear KdV equation.

math.NA↗

arXiv2019

Automatic Variationally Stable Analysis for FE Computations: An Introduction

We introduce an automatic variationally stable analysis (AVS) for finite element (FE) computations of scalar-valued convection-diffusion equations with non-constant and highly oscillatory coefficients. In the spirit of least squares FE methods, the AVS-FE method recasts the governing second order partial differential equation (PDE) into a system of first-order PDEs. However, in the subsequent derivation of the equivalent weak formulation, a Petrov-Galerkin technique is applied by using different regularities for the trial and test function spaces. We use standard FE approximation spaces for the trial spaces, which are C0, and broken Hilbert spaces for the test functions. Thus, we seek to compute pointwise continuous solutions for both the primal variable and its flux (as in least squares FE methods), while the test functions are piecewise discontinuous. To ensure the numerical stability of the subsequent FE discretizations, we apply the philosophy of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan, by invoking test functions that lead to unconditionally stable numerical systems (if the kernel of the underlying differential operator is trivial). In the AVS-FE method, the discontinuous test functions are ascertained per the DPG approach from local, decoupled, and well-posed variational problems, which lead to best approximation properties in terms of the energy norm. We present various 2D numerical verifications, including convection-diffusion problems with highly oscillatory coefficients and extremely high Peclet numbers, up to a billion. These show the unconditional stability without the need for any upwind schemes nor any other artificial numerical stabilization. The results are not highly diffused for convection-dominated problems ...

math.NA↗