High-order Accurate Finite Volume Scheme Research Articles

Imposing slip conditions on curved boundaries for 3D incompressible flows with a very high-order accurate finite volume scheme on polygonal meshes

The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.

Very high-order accurate finite volume scheme for the steady-state incompressible Navier–Stokes equations with polygonal meshes on arbitrary curved boundaries

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence.

Efficient very high-order accurate polyhedral mesh finite volume scheme for 3D conjugate heat transfer problems in curved domains

The conjugate heat transfer problem is found in many engineering applications and comprises the heat transfer between solids and fluids in contact. The discretization method of these problems must guarantee that the interface conditions are properly satisfied, which is an additional difficulty when designing a numerical scheme with very high-order of convergence. The three-dimensional case with arbitrary curved domains is particularly challenging to treat, as the standard approaches to preserve the optimal convergence order become cumbersome with the use of curved meshes and non-linear transformations. Moreover, the implementation aspects of very high-order of convergence methods are critical for the three-dimensional case, as the computational cost of conventional algorithms often grows exponentially with the problem dimension. A very high-order accurate finite volume scheme with general polyhedral meshes is proposed in this work, which is able to solve three-dimensional conjugate heat transfer problems in arbitrary curved domains. The implementation of the proposed method is addressed and optimized calculations are derived to provide the same approximate solution at a significantly reduced computational cost. The provided set of numerical benchmarks, addressing several situations of the conjugate heat transfer problem, allows confirming that the optimal convergence order is effectively achieved. The results obtained also show that the proposed method overcomes the main shortcomings of the standard curved mesh approaches, while preserving the accuracy and the convergence order with the sole use of polyhedral meshes. A computational benchmark proves that substantial performance improvements are obtained from the proposed optimization, which represents a significant advance towards less demanding and more efficient numerical simulations.

Very high-order accurate polygonal mesh finite volume scheme for conjugate heat transfer problems with curved interfaces and imperfect contacts

The conjugate heat transfer problem is found in non-isothermal physical systems that involve thermodynamic processes between materials that are thermally coupled through non-adiabatic contacts. A very high-order accurate finite volume scheme in general polygonal meshes is proposed to solve conjugate heat transfer problems with arbitrary curved interfaces and imperfect thermal contacts. Besides the discontinuous thermal properties, imperfect thermal contacts are challenging to address since the obtained temperature is also discontinuous on the interface as a consequence of the interfacial thermal resistance. Moreover, the arbitrary curved interfaces are discretized with polygonal meshes to avoid the shortcomings of classical curved mesh approaches, but a specific treatment is required to overcome the geometrical mismatch and properly fulfill the prescribed interface conditions. The proposed method is based on a partitioned formulation of the conjugate heat transfer problem with the appropriate thermal coupling. A generic polynomial reconstruction method is used to provide local approximations of the temperature complemented with the reconstruction for off-site data method to properly fulfill the prescribed interface conditions. A comprehensive numerical benchmark is provided to verify the proposed method and assess its capability in terms of accuracy, convergence order, stability, and robustness. The results provide the optimal very high-order of convergence and prove the capability of the method to handle arbitrary curved interfaces and imperfect thermal contacts. This contribution represents a significant step towards more efficient and versatile numerical methods for complex conjugate heat transfer problems in engineering applications.

Very high-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection–diffusion equation with Dirichlet condition

• Very high-order finite volume scheme for curved geometries. • Reconstruction for Off-site Data to provide the optimal order with curved domain. • No curved element and complex mapping on reference element. • A set numerical experiences with non-trivial geometries to assess the method order. Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a very high-order accuracy and the optimal convergence rate is achieved.

Numerical simulations of ultrasound-induced deformation of the lung surface

Ultrasound-induced lung hemorrhage remains the only known bioeffect of non-contrast, diagnostic ultrasound (DUS) found to occur in mammals. However, a fundamental understanding of the ultrasound-lung interaction is still lacking. In this study, we numerically simulate the deformation of the lung surface when subjected to clinically relevant pressure waves. We model the lung as air and the surrounding tissue as water. The two-dimensional, compressible Navier-Stokes equations are solved using a high-order accurate finite volume scheme. We first consider an air-water interface with a small sinusoidal perturbation and study the growth of the perturbations when subjected to a planar waveforms consisting of a step change in pressure. Scaling laws relating the waveform properties to the interface deformation rate are proposed for this simple case. Simulations with DUS waveforms indicate that the interface deformation depends on the unsteady nature of the wave. Preliminary analysis suggests that the initial interface deformation is driven by baroclinic vorticity deposited along the perturbed interface, and that the resulting stresses and strains are a possible candidate causing bioeffects.

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

AbstractIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions witha posterioridetection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.