A sixth-order finite volume method for multidomain convection–diffusion problem with discontinuous coefficients

Machado

International Journal of Applied Mathematics and Computer Science

2015

A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.

Section: Numerical Resultsmentioning

confidence: 99%

“…, I} and φ i be an approximation of the mean value of φ over cell K i . We define the polynomial conservative reconstruction of degree d as (see Clain et al, 2013) …”

Section: Polynomial Reconstruction On Cellsmentioning

confidence: 99%

Section: Polynomial Reconstruction Operatormentioning

confidence: 99%

Section: High-order Fluxesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

Machado

International Journal of Applied Mathematics and Computer Science

2015

Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries

Nóbrega

Numerical Meth Engineering

et al. 2018

Summary Obtaining very high‐order accurate solutions in curved domains is a challenging task as the accuracy of discretization methods may dramatically reduce without an appropriate treatment of boundary conditions. The classical techniques to preserve the nominal convergence order of accuracy, proposed in the context of finite element and finite volume methods, rely on curved mesh elements, which fit curved boundaries. Such techniques often demand sophisticated meshing algorithms, cumbersome quadrature rules for integration, and complex nonlinear transformations to map the curved mesh elements onto the reference polygonal ones. In this regard, the reconstruction for off‐site data method, proposed in the work of Costa et al, provides very high‐order accurate polynomial reconstructions on arbitrary smooth curved boundaries, enabling integration of the governing equations on polygonal mesh elements, and therefore, avoiding the use of complex integration quadrature rules or nonlinear transformations. The method was introduced for Dirichlet boundary conditions and the present article proposes an extension for general boundary conditions, which represents an important advance for real context applications. A generic framework to compute polynomial reconstructions is also developed based on the least‐squares method, which handles general constraints and further improves the algorithm. The proposed methods are applied to solve the convection‐diffusion equation with a finite volume discretization in unstructured meshes. A comprehensive numerical benchmark test suite is provided to verify and assess the accuracy, convergence orders, robustness, and efficiency, which proves that boundary conditions on arbitrary smooth curved boundaries are properly fulfilled and the nominal very high‐order convergence orders are effectively achieved.

High‐order accurate conjugate heat transfer solutions with a finite volume method in anisotropic meshes with application in polymer processing

Nóbrega

Numerical Meth Engineering

et al. 2021

The computational modeling has become an indispensable tool to support the engineering design and control of many industrial processes. In polymer processing applications, the simulation of conjugate heat transfer phenomena is critical as rigorous temperature control is necessary to ensure that the produced parts meet the required specifications. In this article, a high-order accurate finite volume method is proposed to improve the numerical accuracy and the computational efficiency of conjugate heat transfer simulations with application in polymer processing. Anisotropic meshes are investigated to significantly reduce the number of unknowns in convection-dominated problems where the higher temperature variations occur perpendicularly to curved boundaries and interfaces. The reconstruction for off-site data method based on polynomial reconstructions is employed to fulfill the prescribed boundary and interface conditions solely using polygonal meshes to avoid the limitations of curved mesh approaches. A code verification benchmark based on manufactured solutions proves that the proposed method provides a fourth-order of convergence and is computationally more cost-effective than the classical second-order of convergence. Moreover, meshes with higher aspect ratios improve the calculation efficiency but suffer from a small accuracy penalty due to conditioning deterioration. Comparison with the classical finite volume discretization techniques, widely implemented in commercial and open-source software packages, is also provided. The applicability and performance of the proposed method are further supported with a practical case study for the sheet extrusion cooling stage. K E Y W O R D Sanisotropic polygonal meshes, conjugate heat transfer problems, curved boundaries and interfaces, high-order accurate finite volume method, OpenFOAM® software, polymer processing applications 1146