Three-Dimensional Aeroelastic Solutions via the Nonlinear Frequency-Domain Method

Tardif, Pierre-Olivier; Nadarajah, Siva

doi:10.2514/1.j054849

Cited by 16 publications

(9 citation statements)

References 95 publications

(121 reference statements)

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“…The deformation of the mesh is performed using the Radial Basis Functions (RBF) [8]. The method is based on the assumption that the movement of all grid points can be interpolated from the a priori known motion of a set of points called the RBF points.…”

Section: Dynamic Mesh Deformation Using the Radial Basis Functionmentioning

confidence: 99%

“…In our case, it yields a hexahedral stuctured finite-volume framework and a temporal discretization using the NLFD method. A first approach to enforce the VCL in the NLFD context was presented by Tardiff et al [8] but more investigation is needed to determine its limitations. In this section, further developments are added to this approach which expose its analytical limits and a new method is proposed.…”

Section: Derivation Of the Gcl In The Nlfd Frameworkmentioning

confidence: 99%

“…The NLFD method was used to develop a two dimensional aeroelastic flow solver [7] and then expanded to three-dimensional cases on moving grids [8]. In order to study aeroelastic problems, the mesh is moved in time and additional care has to be taken to compute the mesh velocities and the metrics from the physical to the computational space.…”

Section: Introductionmentioning

confidence: 99%

“…The framework of our study is based on the previous development of a three dimensional aeroelastic solver by Tardiff et al [8] who introduced a methodology to enforce the GCL. The expression of the governing equations according to an Arbitrary Lagrangian-Eulerian formulation, the principles of the NLFD discretization and the dynamic mesh deformation using Radial Basis Functions are presented in section 2.…”

Section: Introductionmentioning

confidence: 99%

“…The expression of the governing equations according to an Arbitrary Lagrangian-Eulerian formulation, the principles of the NLFD discretization and the dynamic mesh deformation using Radial Basis Functions are presented in section 2. Additional analytical developments are introduced to demonstrate the limitations of the approach of Tardiff et al [8]. A modified method is derived to satisfy the GCL by computing the integrated face mesh velocities at a second-order accuracy in time.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the Geometric Conservation Law for the Non Linear Frequency Domain and Time-Spectral methods

Benoit

Nadarajah

2019

Computer Methods in Applied Mechanics and Engineering

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The aim of this paper is to present and validate two new procedures to enforce the Geometric Conservation Law (GCL) on a moving grid for an Arbitrary Lagrangian Eulerian (ALE) formulation of the Euler equations discretized in time for either the Non Linear Frequency Domain (NLFD) or Time-Spectral (TS) methods. The equations are spatially discretized by a structured finite-volume scheme on a hexahedral mesh. The derived methodologies follow a general approach where the positions and the velocities of the grid points are known at each time step. The integrated face mesh velocities are derived either from the Approximation of the Exact Volumetric Increments (AEVI) relative to the undeformed mesh or exactly computed based on a Trilinear Mapping (TRI-MAP) between the physical space and the computational domain. The accuracy of the AEVI method highly depends on the computation of the volumetric increments and limits the temporalorder of accuracy of the deduced integrated face mesh velocities to between one and two. Thus defeating the purpose of the NLFD method which possesses spectral rate of convergence. However, the TRI-MAP method has proven to be more computationally efficient, ensuring the satisfaction of the GCL once the convergence of the time derivative of the cell volume is reached in Fourier space. The methods are validated numerically by verifying the conservation of uniform flow and by comparing the integrated face mesh velocities to the exact values derived from the mapping. a condition limits the inherent spectral in time accuracy of the NLFD approach, an alternative method based on a trilinear mapping between the physical and computational space is introduced.The results are then extent to the Time-Spectral method. These various approaches are presented section 3. The different methodologies are numerically investigated in order to verify that the GCL are enforced through a correct computation of the integrated face mesh velocities in section 4 and 50 the results discussed in section 5. Discretization of the governing equation and mesh deformationThis section presents the formulation of the Euler equations on a moving mesh using the Arbitrary Lagrangian-Eulerian approach, its discretization using the Non-Linear Frequency Domain, and the mesh deformation method through the Radial Basis Functions. 55 Arbitrary Lagrangian-Eulerian formulation of the Euler equationsWhen solving the Euler equations on a moving grid a popular approach is to use an Arbitrary Lagrangian-Eulerian (ALE) formulation [14]. For a control volume Ω enclosed by a boundary ∂Ω and without source terms, the integral form of this formulation is given as follows :by the following corollary 3.6 :

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Section: Dynamic Mesh Deformation Using the Radial Basis Functionmentioning

confidence: 99%

Section: Derivation Of the Gcl In The Nlfd Frameworkmentioning

confidence: 99%