Moving Mesh Strategy Based on a Gradient Flow Equation for Two-Dimensional Problems

1998 ◽  
Vol 20 (3) ◽  
pp. 998-1015 ◽  
Author(s):  
Weizhang Huang ◽  
Robert D. Russell
Author(s):  
Alexander Mielke

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.


2006 ◽  
Vol 37 (5) ◽  
pp. 1657-1687 ◽  
Author(s):  
G. Bellettini ◽  
M. Novaga ◽  
E. Paolini

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Dmitri Bykov ◽  
Dieter Lüst

AbstractIt is shown that the Pohlmeyer map of a $$\sigma $$ σ -model with a toric two-dimensional target space naturally leads to the ‘sausage’ metric. We then elaborate the trigonometric deformation of the $$\mathbb {CP}^{n-1}$$ CP n - 1 -model, proving that its T-dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold $$\sigma $$ σ -models and Toda field theories.


2001 ◽  
Vol 16 (11) ◽  
pp. 1889-1898
Author(s):  
WALTER METZNER

We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we compute the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.


Author(s):  
Richard B. Medvitz ◽  
Michael L. Jonson ◽  
James J. Dreyer ◽  
Jarlath McEntee

High resolution RANS CFD analysis is performed to support the design and development of the Ocean Renewable Power Company (ORPC) TidGen™ multi-directional tidal turbine. Two-dimensional and three-dimensional unsteady, moving-mesh CFD is utilized to parameterize the device performance and to provide guidance for device efficiency improvements. The unsteady CFD analysis was performed using a well validated, naval hydrodynamic CFD solver and implementing dynamic overset meshes to perform the relative motion between geometric components. This dynamic capability along with the turbulence model for the expected massively separated flows was validated against published data of a high angle of attack pitching airfoil. Two-dimensional analyses were performed to assess both blade shape and operating conditions. The blade shape performance was parameterized on both blade camber and trailing edge thickness. The blades shapes were found to produce nearly the same power generation at the peak efficiency tip speed ratio (TSR), however off-design conditions were found to exhibit a strong dependency on blade shape. Turbine blades with the camber pointing outward radially were found to perform best over the widest range of TSR’s. In addition, a thickened blade trailing edge was found to be superior at the highest TSR’s with little performance degradation at low TSR’s. Three-dimensional moving mesh analyses were performed on the rotating portion of the full TidGen™ geometry and on a turbine blade stack-up. Partitioning the 3D blades axially showed that no sections reached the idealized 2D performance. The 3D efficiency dropped by approximately 12 percentage points at the peak efficiency TSR. A blade stack-up analysis was performed on the complex 3D/barreled/twisted turbine blade. The analysis first assessed the infinite length blade performance, next end effects were introduced by extruding the 2D foil to the nominal 5.6m length, next barreling was added to the straight foils, and finally twist was added to the foils to reproduce the TidGen™ geometry. The study showed that making the blades a finite length had a large negative impact on the performance, whereas barreling and twisting the foils had only minor impacts. Based on the 3D simulations the largest factor impacting performance in the 3D turbine was a reduction in mass flow through the turbine due to the streamlines being forces outward in the horizontal plane due to the turbine flow resistance. Strategies to mitigate these 3D losses were investigated, including adding flow deflectors on the turbine support structure and stacking multiple turbines in-line.


2012 ◽  
Vol 11 (1) ◽  
pp. 114-146 ◽  
Author(s):  
Peng He ◽  
Huazhong Tang

AbstractThis paper extends the adaptive moving mesh method developed by Tang and Tang [36] to two-dimensional (2D) relativistic hydrodynamic (RHD) equations. The algorithm consists of two “independent” parts: the time evolution of the RHD equations and the (static) mesh iteration redistribution. In the first part, the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations. The second part is an iterative procedure. In each iteration, the mesh points are first redistributed, and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.


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