Lines Matching refs:a

3 HONEE solves the time-dependent Navier-Stokes equations of compressible gas dynamics in a static Eu…
20 …nergy), $\bm{I}_3$ represents the $3 \times 3$ identity matrix, $\bm{b}$ is a body force vector (e…
85 To obtain a finite element discretization, we first multiply the strong form {eq}`eq-vector-ns` by …
91 …}(\Omega_e) \,|\, \bm v(\bm x_e(\bm X)) \in P_p(\bm{I}), e=1,\ldots,N_e \}$ a mapped space of poly…
104 where $\bm{F}(\bm q_N) \cdot \widehat{\bm{n}}$ is typically replaced with a boundary condition.
107 …\bm F$ represents contraction over both fields and spatial dimensions while a single dot represent…
157 Each nonlinear system {eq}`eq-ts-implicit-ns` will correspond to a weak form, as explained below.
158 In determining how difficult a given problem is to solve, we consider the Jacobian of {eq}`eq-ts-im…
164 … by the second term, which is a sort of "mass matrix", and typically well-conditioned independent …
165 In contrast, the first term dominates for large time steps, with a condition number that grows with…
171 We solve {eq}`eq-weak-vector-ns` using a Galerkin discretization (default) or a stabilized method, …
174 Our formulation follows {cite}`hughesetal2010`, which offers a comprehensive review of stabilizatio…
196 …This method is a simplified version of *SUPG* {eq}`eq-weak-vector-ns-supg` which is developed for …
211 …ubgrid state fluctuations $\tilde{\bm q} = -\bm\tau \bm r$ where $\bm r$ is a strong form residual.
229 To build intuition, consider a boundary layer element of dimension $(1, \epsilon)$, for which $\nab…
230 So a small normal component of velocity will be amplified (by a factor of the aspect ratio $1/\epsi…
231 The ratio $\lVert \bm u \rVert / \lVert \bm u_{\bm X} \rVert$ is a covariant measure of (half) the …
232 A contravariant measure of element length in the direction of a unit vector $\hat{\bm n}$ is given …
234 If we consider a parallelogram, the covariant measure is larger than the contravariant measure for …
243 For scalar advection-diffusion, the stabilization is a scalar
257 where the term in parentheses is a rank-1 diffusivity tensor that has been pulled back to the refer…
262 For the Navier-Stokes and Euler equations, {cite}`whiting2003hierarchical` defines a $5\times 5$ di…
289 For Advection-Diffusion, we first examine a 1D definition given by:
296 To make this definition compatible with higher dimensional domains, we use a similar system from th…
306 Note that $\bm g$ is scaled so that it is identity for a unit square, keeping this definition align…
310 In the Euler code, we follow {cite}`hughesetal2010` in defining a $3\times 3$ diagonal stabilizatio…
316 where $c_{\tau}$ is a multiplicative constant reported to be optimal at 0.5 for linear elements, $\…
317 …{\text{adv}}}{\partial \bm q} \cdot \hat{\bm n}_i$ in each direction $i$ is a $5\times 5$ matrix w…
321 \Lambda_i = [u_i - a, u_i, u_i, u_i, u_i+a],
324 where $u_i = \bm u \cdot \hat{\bm n}_i$ is the velocity component in direction $i$ and $a = \sqrt{\…
325 …ear acoustic waves while the middle three are linearly degenerate, carrying a contact wave (temper…
329 …}} \Bigl( \frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \cdot \hat{\bm n}_i \Bigr) = |u_i| + a
332 … as $\gamma$ is an ideal gas parameter; other equations of state will yield a different acoustic w…
337 {ref}`problem-advection`, the problem of the transport of energy in a uniform vector velocity field…
342 This term requires a second derivative to evaluate; first to evaluate $\bm \sigma$ and $\nabla T$ f…
345 To circumvent these issues, we (optionally) perform a projection operation to get $\nabla \cdot \bm…
355 …g 12 scalars-per-node: 4 conserved scalars (mass conservation does not have a diffusive flux term)…
361 Then $\nabla \cdot \bm F_{\text{diff}}$ itself is a function on the finite element space and can be…
369 …ff}}(\bm{q}_N)$ at quadrature points, so we apply integration-by-parts to achieve a calculable RHS:
388 When a fluid simulation is under-resolved (the smallest length scale resolved by the grid is much l…
412 For explicit LES, it is defined by a subgrid stress model.
431 The data-driven SGS model implemented here uses a small neural network to compute the SGS term.
437 The outputs of the network are assumed to be normalized on a min-max scale, so they must be rescale…
438 Parameters for the neural network are put into files in a directory found in `-sgs_model_dd_paramet…
452 In fused mode, the input processing, model inference, and output handling were all done in a single…
453 …r fused mode requires that the model architecture be manually implemented into a libCEED QFunction.
464 Note that if you chose to run the inference on host while using a GPU libCEED backend (e.g. `/gpu/c…
466 The sequential mode is available using a libCEED based inference evaluation via `-sgs_model_dd_impl…
510 Below follows a re-description of the formulation to match the present notation, and then a descrip…
534 The set of wavemode frequencies is defined by a geometric distribution:
540 The wavemode amplitudes $q^n$ are defined by a model energy spectrum $E(\kappa)$:
554 …pproximates the effective cutoff frequency of the mesh (viewing the mesh as a filter on solution o…
613 The `STGInflow.dat` file is a table of values at given distances from the wall.
614 These values are then interpolated to a physical location (node or quadrature point). It has the fo…
619 where each `[  ]` item is a number in scientific notation (ie. `3.1415E0`), and `sclr_1` and `sclr_…
711 IDL is not a boundary condition, but it's primary application is for use with STG.
715 …d by {cite}`shurSTG2014`, but is implemented here as a ramped volumetric forcing term, similar to …
722 where $\bm{Y}' = [P - P_\mathrm{ref}, \bm{0}, 0]^T$, and $\sigma(\bm{x})$ is a linear ramp starting…
723 The damping is defined in terms of a pressure-primitive anomaly $\bm Y'$ converted to conservative …
724 $P_\mathrm{ref}$ has a default value equal to `-reference_pressure` flag, with an optional flag `-i…