The finite-volume method in computational rheology

The finite volume method (FVM) is widely used in traditional computational fluid dynamics (CFD), and many commercial CFD codes are based on this technique which is typically less demanding in computational resources than finite element methods (FEM). However, for historical reasons, a large number of Computational Rheology codes are based on FEM. There is no clear reason why the FVM should not be as successful as finite element based techniques in Computational Rheology and its applications, such as polymer processing or, more recently, microfluidic systems using complex fluids. This chapter describes the major advances on this topic since its inception in the early 1990’s, and is organized as follows. In the next section, a review of the major contributions to computational rheology using finite volume techniques is carried out, followed by a detailed explanation of the methodology developed by the authors. This section includes recent developments and methodologies related to the description of the viscoelastic constitutive equations used to alleviate the high-Weissenberg number problem, such as the log-conformation formulation and the recent kernel-conformation technique. At the end, results of numerical calculations are presented for the well-known benchmark flow in a 4:1 planar contraction to ascertain the quality of the predictions by this method

Lead extrusion analysis by finite volume method

Computational numerical simulation is nowadays largely applied in the design and analysis of metal forming process. Extrusion of metals is one main forming process largely applied in the manufacturing of metallic products or parts. Historically, the Finite Element Method has been applied for decades in metal extrusion analysis [4]. However, recently in the academy, there is a trend to use Finite Volume Method: literature suggests that metal flow by extrusion can be analyzed by the flow formulation [1, 2]. Thus, metal flow can be modelled such us an incompressible viscous fluid [2]. This hypothesis can be assumed because extrusion process is an isochoric process. The MacCormack Method is commonly used to simulate compressible fluid flow by the finite volume method [3]. However, metal extrusion and incompressible fluid flow do not present state equations for the evolution of pressure, and therefore, a velocity-pressure coupling method is necessary to obtain a consistent velocity and pressure fields [3]. Present work proposes a new numerical scheme to obtain information about metal flow in the extrusion process, in steady state. The governing equations were discretized by Finite Volume Method, using the Explicit MacCormack Method to structured and collocated mesh. The SIMPLE Method was applied to attain pressure-velocity coupling [3]. These new numerical scheme was applied to forward extrusion process of lead. The incompressible metal extrusion velocity fields achieved faster convergence and a good agreement with analytical and experimental results obtained from literature. The MacCormack Method applied for metals produced consistent results without the need of artificial viscosity as employed by the compressible flow simulation approaches. Furthermore, the present numerical results also suggest that MacCormack Method and SIMPLE can be applied in the solution of metal forming processes besides the traditional application for compressible fluid flow

Aluminium extrusion analysis by the finite volume method

Present work proposes a novel numerical scheme to calculate stress and velocity fields of metal flow in axisymmetric extrusion process in steady state. Extrusion of aluminium is one main metal forming process largely applied in manufacturing bars and products with complex cross section shape. The upper-bound, slab, slip-line methods and more recently the numerical methods such as the Finite Element Method have been commonly applied in aluminium extrusion analysis. However, recently in the academy, the Finite Volume Method has been developed for metal flow analysis: literature suggests that extrusion of metals can be modelled by the flow formulation. Hence, metal flow can be mathematically modelled such us an incompressible non linear viscous fluid, owing to volume constancy and varying viscosity in metal forming. The governing equations were discretized by the Finite Volume Method, using the Explicit MacCormack Method in structured and collocated mesh. The MacCormack Method is commonly used to simulate compressible fluid flow by the finite volume method. However, metal plastic flow and incompressible fluid flow do not present state equations for the evolution of pressure, and therefore, a velocity-pressure coupling method is necessary to obtain a consistent velocity and pressure fields. The SIMPLE Method was applied to attain pressure-velocity coupling. This new numerical scheme was applied to forward hot extrusion process of an aluminium alloy. The metal extrusion velocity fields achieved fast convergence and a good agreement with experimental results. The MacCormack Method applied to metal extrusion produced consistent results without the need of artificial viscosity as employed by the compressible flow simulation approaches. Therefore, present numerical results also suggest that MacCormack method together with SIMPLE method can be applied in the solution of metal forming processes in addition to the traditional application for compressible fluid flow

Hybrid Spectral Difference/Embedded Finite Volume Method for Conservation Laws

A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field. The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.Comment: 27 pages, 17 figures, 2 tables, Accepted for publication in the Journal of Computational Physics, April 201