High Order Finite Research Articles

An Adaptive High-Order Surface Finite Element Method for Polymeric Self-Consistent Field Theory on General Curved Surfaces

An Adaptive High-Order Surface Finite Element Method for Polymeric Self-Consistent Field Theory on General Curved Surfaces

KFVM-WENO: A High-order Accurate Kernel-based Finite Volume Method for Compressible Hydrodynamics

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided.

Comparison of high-order numerical methodologies for the simulation of the supersonic Taylor–Green vortex flow

This work presents a comparison of several high-order numerical methodologies for simulating shock/turbulence interactions based on the supersonic Taylor–Green vortex flow, considering a Reynolds number of 1600 and a Mach number of 1.25. The numerical schemes considered include high-order Finite Difference, Targeted Essentially Non-Oscillatory, Discontinuous Galerkin, and Spectral Difference schemes. The shock capturing methods include high-order filtering, localized artificial diffusivity, non-oscillatory numerical fluxes, and local low-order switching. The ability of the various high-order numerical methodologies to both capture shocks and represent accurately the development of turbulent vortices is assessed.

Combination of Advanced Actuator Line/Disk Model and High-Order Unstructured Finite Volume Solver for Helicopter Rotors

In the research field of rotorcraft aerodynamics, there are two fundamental challenges: resolving the complex vortex structures in rotor wakes and representing the moving rotor blades in the ambient airflow. In this paper, we address the first challenge by utilizing a third-order unstructured finite volume solver, which exhibits lower numerical dissipation than its second-order counterpart. This allows for sufficient resolution of small vortex structures on relatively coarse meshes. With this flow solver, the second challenge is addressed by modeling each rotor as an actuator disk (i.e., the actuator disk model (ADM)) or modeling each blade as an actuator line (i.e., the actuator line model (ALM)). Both of the two models are equipped with an improved tip loss correction, which is introduced in detail in the methodology section. In the section of numerical experiments, the numerical convergence properties of the two types of solvers have been compared in the case of two-dimensional infinite wing. In addition, the relationship between the ALM and the lifting line theory is discussed in the cases of fixed-wing calculations. Another goal of these cases is to validate the tip loss correction presented. The validation of the ALM/ADM and comparisons of computational efficiency are also demonstrated in simulations involving both hover and forward flight rotors. It was found that the combination of the third-order finite volume solver and the ALM/ADM with the improved tip loss correction presents an efficient way of performing the aerodynamic analysis of rotor-induced downwash flow.

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

High-Order Well-Balanced Finite Difference AWENO Schemes for Shallow Water Flows along Channels with Irregular Geometry

High-Order Well-Balanced Finite Difference AWENO Schemes for Shallow Water Flows along Channels with Irregular Geometry

New procedure for soil-structure interaction problems incorporating free-field effects by 2D quadratic elements

Following several decades of tragic seismic events, current national regulations emphasise the importance of considering seismic soil-structure interaction (SSSI) in the design of foundations, particularly for deep foundations in loose soils. However, incorporating the free field into the SSSI calculation model can be challenging when high order finite or infinite elements are involved.This paper presents a new procedure for considering the effect of the free field in a seismic soil-structure interaction (SSSI) model when using 8-node finite elements bounded by compatible absorbing infinite elements.The proposal, based on a new formulation of three-node column elements paired with a modification of the Cundall method, allows the calculation of the free field effects to be applied at the level of the lateral boundaries of the SSSI model, represented here by the finite-infinite element interfaces. This modification of the dissipative system, considering the kinematic effect at each time step, allows an incremental solution of the SSSI problem, taking into account the nonlinearities of the materials constituting the geotechnical profile. The test validation of the method, performed on two cases of shear wave propagation problems in soft soils, shows good results, demonstrating the usefulness of the 2D approach for SSSI problems.

Structural Behavior of Water Flow Glazing: Stress and Elastic Deformation Considering Hydrostatic Pressure

The fluid inside a Water Flow Glazing (WFG) panel creates a linear pressure distribution along the vertical dimension. Tension stress can cause problems with the sealant; compression stress can cause deflections in the glass panel. Increasing the glass thickness until the deflection is below the required limit implies more weight and cost of the glazing. Another solution is to limit glass deflection by inserting pillars or stripes into the cavity between two glass panes. The novelty of this article was to test a High-Order Finite Difference Method for linear and non-linear models to evaluate the effect of hydrostatic pressure produced by the fluid chamber on WFG panels. The methodology was tested on two case studies to assess the tension and deflection of the glass panes to guarantee the structural stability of WFG. The main conclusion drawn was that a linear plate model was sufficient to dimension the width, length, and thickness of the WFG panel. Furthermore, the mathematical model provided criteria to keep the glass tension below 45 MPa for tempered glass and the maximum deflection as the minimum between 1‰ of the glass height and 10% of the water chamber thickness. Introducing pillars or stripes solved hydrostatic pressure problems when the panel’s height was above 1.5 m.

The Compact and Accuracy Preserving Limiter for High-Order Finite Volume Schemes Solving Compressible Flows

The Compact and Accuracy Preserving Limiter for High-Order Finite Volume Schemes Solving Compressible Flows

Free Vibration and Buckling Analysis of Porous Two-Directional Functionally Graded Beams Using a Higher-Order Finite Element Model

Free Vibration and Buckling Analysis of Porous Two-Directional Functionally Graded Beams Using a Higher-Order Finite Element Model

Heat and mass transfer characteristics in MHD Casson fluid flow over a cylinder in a wavy channel: Higher-order FEM computations

In this paper, an extensive analysis of heat and mass transfer characteristics of a Casson fluid flow past a cylinder in a wavy channel has been performed. Due to the non-availability of analytical solution the model represented by coupled non-linear partial differential equations has been simulated by implementing higher-order Finite Element Method (FEM). For the approximation of velocity, temperature, and concentration profiles, a finite element space involving the cubic polynomial (P3) is selected whereas the pressure estimation is accomplished through a space of quadratic polynomial (P2). Such a choice gives rise to 10° of freedom for each of the velocity components, temperature, and concentration profile while 6° of freedom for the pressure. The Newton's method is used to linearize the systems of equations. The linearized inner system of equations are solved with a direct solver PARDISO. Computational results are demonstrated in the form of velocity, isotherms, isoconcentrations, and for several quantities of interest including the drag and lift coefficients. It has been observed that both drag and lift coefficients show a decreasing trend with Magnetic parater M. In addition, for all values of Bn, the drag coefficient has a linear growth profile as a result of fluid forces dominating at higher values of Bn. Furthermore, the concentration is high in the center of the channel and has a decreasing trend as we move towards the walls.

PROGRESSIVE DAMAGE ANALYSIS OF STEEL-REINFORCED CONCRETE BEAMS USING HIGHER-ORDER 1D FINITE ELEMENTS

The present work investigates progressive damage in steel-reinforced concrete structures. An elastic-perfectly plastic material response is considered for the reinforcing steel constituent, while the smeared-crack approach is applied to model the nonlinear behavior of concrete. The analysis employs one-dimensional numerical models based on higher-order finite elements derived using the Carrera unified formulation (CUF). A set of numerical assessments is presented to study the mechanical response of a steel-reinforced notched concrete beam loaded in tension. The predictions are found to be in very good agreement with reference experimental observations, thereby validating the numerical approach. It is shown that CUF allows for the explicit representation of the constituents within the composite beam, resulting in accurate solutions in a computationally efficient manner.

Three-Dimensional Modeling of Marine Controlled Source Electromagnetic Using a New High-Order Finite Element Method With an Absorption Boundary Condition

The marine controlled source electromagnetic method (MCSEM) is widely used in marine oil, gas exploration and deep structures investigation because of its low cost and high efficiency. In this paper, a high-precision forward algorithm for 3D modeling for MCSEM is proposed. Unstructured meshes are used to discretize the model domain, and local mesh refinement techniques are applied, which is conducive to simulating complex terrains and targets. The application of electric dipole discrete technology can load wire excitation sources with arbitrarily complex spatial shapes, which can simulate more realistic electromagnetic field distribution of marine controllable sources electromagnetic in actual exploration. Absorption boundary conditions based on real number and exponential stretching techniques are introduced to improve the numerical solution accuracy. To improve the efficiency of solving, the open-source massively parallel solver (MUMPS) based on the multifrontal algorithm is used to solve the finite element equations. Finally, some typical geoelectric models are designed to verify the correctness and effectiveness of the proposed algorithm. The calculation results show that higher order finite elements bring about higher accuracy. In addition, the loading of absorption boundary conditions is simple and effective than conventional Dirichlet boundary conditions.

High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

A Uniform Accuracy High-Order Finite Difference and FEM for Optimal Problem Governed by Time-Fractional Diffusion Equation

In this paper, the time fractional diffusion equations optimal control problem is solved by 3−α order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the 3−α order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and ‘first optimize, then discretize’ approach. The fully discrete scheme’s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct.

A Well-Conditioned Weak Coupling of Boundary Element and High-Order Finite Element Methods for Time-Harmonic Electromagnetic Scattering by Inhomogeneous Objects

The aim of this paper is to propose efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a non-overlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Pade approximants of the nonlocal Magnetic-to-Electric operators. Numerical results are presented to validate and analyze the new approach for both homogeneous and inhomogeneous scatterers.

An Experimental and Theoretical Comparison of 3D Models for Ultrasonic Non-Destructive Testing of Cracks: Part I, Embedded Cracks

Ultrasonic Non-Destructive Testing (NDT) methods are broadly used for detection and characterization/imaging of cracks. Simulation is of great interest for designing such NDT methods. To model the ultrasonic 3D response of a crack, ultrasonic high frequency asymptotic (semi-analytical) models (such as the Physical Theory of Diffraction—PTD) are known to provide accurate predictions for most classical NDT configurations, and 3D numerical models have also emerged more recently. The aim of this paper is to carry out for the first time an experimental and theoretical comparison of 3D models for ultrasonic NDT of embedded cracks in 3D configurations. Semi-analytical models and a hybrid 3D FEM strategy—combining high-order spectral Finite Elements Method (FEM) for flaw scattering and an asymptotic ray model for beam propagation—have been compared. Both numerical validations and comparisons between simulation and experiments prove the effectiveness of PTD in numerous configurations but validate and demonstrate the improvement provided by the 3D hybrid code, notably for small flaws compared to the wavelength and for shear waves.

High order Finite Difference/Discontinuous Galerkin schemes for the incompressible Navier-Stokes equations with implicit viscosity

Abstract In this work we propose a novel numerical method for the solution of the incompressible Navier-Stokes equations on Cartesian meshes in 3D. The semi-discrete scheme is based on an explicit discretization of the nonlinear convective flux tensor and an implicit treatment of the pressure gradient and viscous terms. In this way, the momentum equation is formally substituted into the divergence-free constraint, thus obtaining an elliptic equation on the pressure which eventually maintains at the discrete level the involution on the divergence of the velocity field imposed by the governing equations. This makes our method belonging to the class of so-called structure-preserving schemes. High order of accuracy in space is achieved using an efficient CWENO reconstruction operator that is exploited to devise a conservative finite difference scheme for the convective terms. Implicit central finite differences are used to remove the numerical dissipation in the pressure gradient discretization. To avoid the severe time step limitation induced by the viscous eigenvalues related to the parabolic terms in the governing equations, we propose to devise an implicit local discontinuous Galerkin (DG) solver. The resulting viscous sub-system is symmetric and positive definite, therefore it can be efficiently solved at the aid of a matrix-free conjugate gradient method. High order in time is granted by a semi-implicit IMEX time stepping technique. Convergence rates up to third order of accuracy in space and time are proven, and a suite of academic benchmarks is shown in order to demonstrate the robustness and the validity of the novel schemes, especially in the context of high viscosity coefficients.

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations