Lines Matching refs:in

3 …in a static Eulerian three-dimensional frame using unstructured high-order finite/spectral element…
8 The mathematical formulation (from {cite}`shakib1991femcfd`) is given in what follows.
9 The compressible Navier-Stokes equations in conservative form are
28 The system {eq}`eq-ns` can be rewritten in vector form
82 with $P=p+1$ the number of nodes in the element $e$.
85 …rst multiply the strong form {eq}`eq-vector-ns` by a test function $\bm v \in H^1(\Omega)$ and int…
88 …dot \bm{F}(\bm{q}_N) - \bm{S}(\bm{q}_N) \right) \,dV = 0 \, , \; \forall \bm v \in \mathcal{V}_p\,,
91 …in H^{1}(\Omega_e) \,|\, \bm v(\bm x_e(\bm X)) \in P_p(\bm{I}), e=1,\ldots,N_e \}$ a mapped space …
100   &= 0 \, , \; \forall \bm v \in \mathcal{V}_p \,,
107 …th fields and spatial dimensions while a single dot represents contraction in just one, which shou…
111 <!-- TODO: This should be reframed in terms of PETSc TS's F(t, u, \dot u) = G(t, u) rather than spe…
117 …rg (RKF4-5) method by default (any explicit time-stepping scheme available in PETSc can be chosen …
143 … method by default (similarly, any implicit time-stepping scheme available in PETSc can be chosen …
156 in terms of $\bm z_N$ from prior state and $\alpha > 0$, both of which depend on the specific time …
164 …scales inversely with the time step $\Delta t$, so small time steps result in the Jacobian being d…
168 More details of PETSc's time stepping solvers can be found in the [TS User Guide](https://petsc.org…
173 … as the Euler equations and (low-viscosity/poorly resolved) Navier-Stokes, in which case stabiliza…
188   \, , \; \forall \bm v \in \mathcal{V}_p
204   & = 0 \, , \; \forall \bm v \in \mathcal{V}_p
210 In both {eq}`eq-weak-vector-ns-su` and {eq}`eq-weak-vector-ns-supg`, $\bm\tau \in \mathbb R^{5\time…
230 …f velocity will be amplified (by a factor of the aspect ratio $1/\epsilon$) in this transformation.
231 … \bm u_{\bm X} \rVert$ is a covariant measure of (half) the element length in the direction of the…
232 A contravariant measure of element length in the direction of a unit vector $\hat{\bm n}$ is given …
233 … readily computable, its inverse $\nabla_{\bm x} \bm X$ is needed directly in finite element metho…
250 …te that $\tau$ has units of time and, in the transport-dominated limit, is proportional to element…
257 where the term in parentheses is a rank-1 diffusivity tensor that has been pulled back to the refer…
267 The Navier-Stokes code in this example uses the following formulation for $\tau_c$, $\tau_m$, $\tau…
285 This formulation is currently not available in the Euler code.
297 This results in the following definition:
310 In the Euler code, we follow {cite}`hughesetal2010` in defining a $3\times 3$ diagonal stabilizatio…
316 …t{\bm n}_i$ is a unit vector in direction $i$, and $\nabla_{x_i} = \hat{\bm n}_i \cdot \nabla_{\bm…
317 The flux Jacobian $\frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \cdot \hat{\bm n}_i$ in each …
318 The complete set of eigenvalues of the Euler flux Jacobian in direction $i$ are (e.g., {cite}`toro2…
324 where $u_i = \bm u \cdot \hat{\bm n}_i$ is the velocity component in direction $i$ and $a = \sqrt{\…
326 The fastest wave speed in direction $i$ is thus
337 {ref}`problem-advection`, the problem of the transport of energy in a uniform vector velocity field…
341 The strong residual in the SUPG operator in {eq}`eq-weak-vector-ns-supg` and {eq}`eq-weak-vector-ns…
346 This was first proposed in {cite}`jansenDiffFluxProjection1999`.
347 There are two methods of achieving this implemented in HONEE, denoted as the direct and indirect me…
351 Indirect projection is the method presented in {cite}`jansenDiffFluxProjection1999`.
355 … conserved scalars (mass conservation does not have a diffusive flux term) in 3 dimensional direct…
383 The $L^2$ projection in either method uses the standard mass matrix, which is rowsum lumped for per…
390 …operation results in an extra stress-like term, $\bm{\tau}^r$, representing the effect of unresolv…
409 More details on deriving the above expression, filtering, and large eddy simulation can be found in…
433 More details regarding the theoretical background of the model can be found in {cite}`prakashDDSGS2…
438 Parameters for the neural network are put into files in a directory found in `-sgs_model_dd_paramet…
440 The first row of each files stores the number of columns and rows in each file.
441 Note that the weight coefficients are assumed to be in column-major order.
445 The data-driven model parameters in the examples directory are not accurate and are for regression …
452 In fused mode, the input processing, model inference, and output handling were all done in a single…
458 …ibraries allows us to leverage the flexibility of those external libraries in their model architec…
509 We use the STG method described in {cite}`shurSTG2014`.
619 where each `[  ]` item is a number in scientific notation (ie. `3.1415E0`), and `sclr_1` and `sclr_…
620 They are not used in this example.
629 The following table is presented to help clarify the dimensionality of the numerous terms in the ST…
699 …- Set the element size in the x direction. Default is calculated for box meshes, assuming equispac…
715 …s a ramped volumetric forcing term, similar to a sponge layer (see 8.4.2.4 in {cite}`colonius2023t…
723 The damping is defined in terms of a pressure-primitive anomaly $\bm Y'$ converted to conservative …
740   - Start of IDL in the x direction
745   - Length of IDL in the positive x direction