PHiLiP::GridRefinement::SizeField< dim, real > Class Template Reference

Size Field Static Class. More...

#include <size_field.h>

Collaboration diagram for PHiLiP::GridRefinement::SizeField< dim, real >:

Static Public Member Functions
static void	isotropic_uniform (const real &complexity, const dealii::Vector< real > &B, const dealii::DoFHandler< dim > &dof_handler, std::unique_ptr< Field< dim, real >> &h_field, const real &poly_degree)
	Computes the size field (element scale) for a uniform \(p\)- distribution. More...
static void	isotropic_h (const real complexity, const dealii::Vector< real > &B, const dealii::DoFHandler< dim > &dof_handler, const dealii::hp::MappingCollection< dim > &mapping_collection, const dealii::hp::FECollection< dim > &fe_collection, const dealii::hp::QCollection< dim > &quadrature_collection, const dealii::UpdateFlags &update_flags, std::unique_ptr< Field< dim, real >> &h_field, const dealii::Vector< real > &p_field)
	Computes the size field (element scale) for a potentially non-uniform \(p\)- distribution. More...
static void	isotropic_hp (const real complexity, const dealii::Vector< real > &Bm, const dealii::Vector< real > &B, const dealii::Vector< real > &Bp, const dealii::DoFHandler< dim > &dof_handler, const dealii::hp::MappingCollection< dim > &mapping_collection, const dealii::hp::FECollection< dim > &fe_collection, const dealii::hp::QCollection< dim > &quadrature_collection, const dealii::UpdateFlags &update_flags, std::unique_ptr< Field< dim, real >> &h_field, dealii::Vector< real > &p_field)
	Computes the size field (element scale) and updated polynomial distrubition for high-order error function. More...
static void	adjoint_uniform_balan (const real complexity, const real r_max, const real c_max, const dealii::Vector< real > &eta, const dealii::DoFHandler< dim > &dof_handler, const dealii::hp::MappingCollection< dim > &mapping_collection, const dealii::hp::FECollection< dim > &fe_collection, const dealii::hp::QCollection< dim > &quadrature_collection, const dealii::UpdateFlags &update_flags, std::unique_ptr< Field< dim, real >> &h_field, const real &poly_degree)
	Performs adjoint based size field adaptation for uniform polynomial distribution based on the method of Balan et al. More...
static void	adjoint_h_balan (const real complexity, const real r_max, const real c_max, const dealii::Vector< real > &eta, const dealii::DoFHandler< dim > &dof_handler, const dealii::hp::MappingCollection< dim > &mapping_collection, const dealii::hp::FECollection< dim > &fe_collection, const dealii::hp::QCollection< dim > &quadrature_collection, const dealii::UpdateFlags &update_flags, std::unique_ptr< Field< dim, real >> &h_field, const dealii::Vector< real > &p_field)
	Performs adjoint based size field adaptation for non-uniform polynomial distribution based on the method of Balan et al. More...
static void	adjoint_h_equal (const real complexity, const dealii::Vector< real > &eta, const dealii::DoFHandler< dim > &dof_handler, const dealii::hp::MappingCollection< dim > &mapping_collection, const dealii::hp::FECollection< dim > &fe_collection, const dealii::hp::QCollection< dim > &quadrature_collection, const dealii::UpdateFlags &update_flags, std::unique_ptr< Field< dim, real >> &h_field, const real &poly_degree)
	Performs adjoint based size field adaptation with a uniform \(p\)- distribution based on equidistribution of error. More...

Static Protected Member Functions
static void	update_h_dwr (const real tau, const dealii::Vector< real > &eta, const dealii::DoFHandler< dim > &dof_handler, std::unique_ptr< Field< dim, real >> &h_field, const real &poly_degree)
	Updates the size field based on the DWR disitribution and a reference value, \(\tau\). More...
static void	update_h_optimal (const real lambda, const dealii::Vector< real > &B, const dealii::DoFHandler< dim > &dof_handler, std::unique_ptr< Field< dim, real >> &h_field, const dealii::Vector< real > &p_field)
	Based on a chosen bisection paraemeter, updates the \(h\)- field to an optimal distribution. More...
static real	evaluate_complexity (const dealii::DoFHandler< dim > &dof_handler, const dealii::hp::MappingCollection< dim > &mapping_collection, const dealii::hp::FECollection< dim > &fe_collection, const dealii::hp::QCollection< dim > &quadrature_collection, const dealii::UpdateFlags &update_flags, const std::unique_ptr< Field< dim, real >> &h_field, const dealii::Vector< real > &p_field)
	Evaluates the continuous complexity (approximate degrees of freedom) for the target \(hp\) mesh representation. More...
static void	update_alpha_vector_balan (const dealii::Vector< real > &eta, const real r_max, const real c_max, const real eta_min, const real eta_max, const real eta_ref, const dealii::DoFHandler< dim > &dof_handler, const dealii::Vector< real > &I_c, std::unique_ptr< Field< dim, real >> &h_field)
	Updates the size field for the mesh through the area measure based on reference value. More...
static real	update_alpha_k_balan (const real eta_k, const real r_max, const real c_max, const real eta_min, const real eta_max, const real eta_ref)
	Determines local \(\alpha\) sizing factor (from adjoint estimates) More...
static real	bisection (const std::function< real(real)> func, real lower_bound, real upper_bound, real rel_tolerance=1e-6, real abs_tolerance=1.0)
	Bisect function based on starting bounds. More...

Detailed Description

template<int dim, typename real>
class PHiLiP::GridRefinement::SizeField< dim, real >

Size Field Static Class.

This class contains various utilities for computing updated continuous mesh representations. Notably, these are used along with continuous error estimation models to drive unstructured mesh adaptation methods from grid_refinement_continuous.cpp. Depending on the remeshing targets, mesh type and information availible, different functions are included for isotropic, anisotropic ($h$-adaptation), varying polynomial order ($h$-adaptation), hybrid ($hp$-adaptation) and goal-oriented (adjoint based) cases. The results are used to modify the local continuous element representation of the Field class (with isotropic or anisotropic element definitions) and a vector of local polynomial orders. For the quad meshing case, this anisotropic field is represented by the frame field:

\[ f_\boldsymbol{x} = \left< \boldsymbol{v}, \boldsymbol{w}, -\boldsymbol{v}, -\boldsymbol{w} \right> = V \left< \boldsymbol{e}_1, \boldsymbol{e}_2, -\boldsymbol{e}_1, -\boldsymbol{e}_2 \right> \]

Where the vectors \(v\) and \(w\) are sized and aligned with the target element axis, and \(V\) is the local transformation matrix from the reference element axis \(e_i\) to form this physical frame. For the orthogonal frame-field adaptation case this can be decomposed as

\[ V = \left[\begin{matrix} \boldsymbol{v} & \boldsymbol{w} \end{matrix}\right] = h \left[\begin{matrix} cos(\thea) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{matrix}\right] \left[\begin{matrix} \sqrt{\rho} & \\ & 1/\sqrt{\rho} \end{matrix}\right] \]

where \(h\) represents the size field, \(\theta\) the orientation and \(\rho\) the anisotropy.Central to many of the techniques is the target complexity defined by

\[ \mathcal{C}(\mathcal{M}) = \int_{\Omega} { \mathrm{det}{(V(\boldsymbol{x}))} \left(\mathcal{P}(\boldsymbol{x})+1\right)^2 \mathrm{d}\boldsymbol{x}} \]

This uses the frame field transformation \(V(x)\) and continuous polynomial field \(\mathcal{P})(x)\). Together these provide an approximation of the continuous degrees of freedoms in the adapted target mesh. Note: \(p-\) based functionality is still not complete and requires updates in other parts of the code.

Definition at line 64 of file size_field.h.

Member Function Documentation

adjoint_h_balan()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::adjoint_h_balan ( const real  complexity, const real  r_max, const real  c_max, const dealii::Vector< real > &  eta, const dealii::DoFHandler< dim > &  dof_handler, const dealii::hp::MappingCollection< dim > &  mapping_collection, const dealii::hp::FECollection< dim > &  fe_collection, const dealii::hp::QCollection< dim > &  quadrature_collection, const dealii::UpdateFlags &  update_flags, std::unique_ptr< Field< dim, real >> &  h_field, const dealii::Vector< real > &  p_field  )

static

Performs adjoint based size field adaptation for non-uniform polynomial distribution based on the method of Balan et al.

Works by scaling the local element sizes according to a logarithmic distribution of the error parameter based on the dual-weighted residual (DWR). This works with non-uniform high-order \(p\)-distrubtions (although they are not modified). It is recommended to use this to determine only the mesh scale update relative to the previous iteration with the anisotropic targets obtained from the directional derivatives quadratic form. Overall, the problem involves iteratively sampling threshold values \(\eta_{ref}\) to drive the choice of refinement/coarsening choices. Care must be taken in setting the complexity target, maximum refinement and maximum coarsening range to ensure that a suitable refinement distribution is maintained throughout grid refinements. See the original paper by Balan et al. for more information "Adjoint-based hp-adaptivity on anisotropic meshes..." and the functions update_alpha_vector_balan and update_alpha_k_balan for implementation details.

Parameters

complexity	Target global continuous mesh complexity
r_max	Maximum refinement scaling factor
c_max	Maximum coarsening scaling factor
eta	Dual-weighted residual (DWR) error indicator distribution
dof_handler	DoFHandler describing the mesh
mapping_collection	Element mapping collection
fe_collection	Finite element collection
quadrature_collection	Quadrature rules collection
update_flags	Update flags for the volume finite elements
h_field	(Output) Target size-field
p_field	polynomial degree vector

Definition at line 307 of file size_field.cpp.

adjoint_h_equal()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::adjoint_h_equal ( const real  complexity, const dealii::Vector< real > &  eta, const dealii::DoFHandler< dim > &  dof_handler, const dealii::hp::MappingCollection< dim > &  mapping_collection, const dealii::hp::FECollection< dim > &  fe_collection, const dealii::hp::QCollection< dim > &  quadrature_collection, const dealii::UpdateFlags &  update_flags, std::unique_ptr< Field< dim, real >> &  h_field, const real &  poly_degree  )

static

Performs adjoint based size field adaptation with a uniform \(p\)- distribution based on equidistribution of error.

This process assumes that the error in the DWR with locally converge at the optimal \(2p+1\) rate and attempts to select a size field distribution which will equally distribute this error and minimize the global error function at a chosen target complexity. Locally this leads to scaling the error relative to some reference bisection parameter with a \(2p+1\) exponent. See update_h_dwr for the description of the size field.

Note: This method is not recommended as it leads to large oscillations in the size field because the error does not predictibly follow this convergence pattern.

Parameters

complexity	Target global continuous mesh complexity
eta	Dual-weighted residual (DWR) error indicator distribution
dof_handler	DoFHandler describing the mesh
mapping_collection	Element mapping collection
fe_collection	Finite element collection
quadrature_collection	Quadrature rules collection
update_flags	Update flags for the volume finite elements
h_field	(Output) Target size-field
poly_degree	(Input) Uniform polynomial degree

Definition at line 419 of file size_field.cpp.

adjoint_uniform_balan()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::adjoint_uniform_balan ( const real  complexity, const real  r_max, const real  c_max, const dealii::Vector< real > &  eta, const dealii::DoFHandler< dim > &  dof_handler, const dealii::hp::MappingCollection< dim > &  mapping_collection, const dealii::hp::FECollection< dim > &  fe_collection, const dealii::hp::QCollection< dim > &  quadrature_collection, const dealii::UpdateFlags &  update_flags, std::unique_ptr< Field< dim, real >> &  h_field, const real &  poly_degree  )

static

Performs adjoint based size field adaptation for uniform polynomial distribution based on the method of Balan et al.

Works by scaling the local element sizes according to a logarithmic distribution of the error parameter based on the dual-weighted residual (DWR). See adjoint_h_balan for details in the more general \(p\)- distribution case.

Parameters

complexity	Target global continuous mesh complexity
r_max	Maximum refinement scaling factor
c_max	Maximum coarsening scaling factor
eta	Dual-weighted residual (DWR) error indicator distribution
dof_handler	DoFHandler describing the mesh
mapping_collection	Element mapping collection
fe_collection	Finite element collection
quadrature_collection	Quadrature rules collection
update_flags	Update flags for the volume finite elements
h_field	(Output) Target size-field
poly_degree	uniform polynomial degree

Definition at line 272 of file size_field.cpp.

bisection()

template<int dim, typename real >

real PHiLiP::GridRefinement::SizeField< dim, real >::bisection ( const std::function< real(real)>  func, real  lower_bound, real  upper_bound, real  rel_tolerance = 1e-6, real  abs_tolerance = 1.0  )

staticprotected

Bisect function based on starting bounds.

Performs bisection on an input lambda function \(f(x)\) with starting bounds \(x\in\left[a,b\right]\). Assumes that \(f(a)\) and \(f(b)\) are of opposite sign and iteratively chooses half interval until a suitably accurate approximation of \(f(x)\approx 0\) is found. Stops when either a absolute or relative (to the initial range) function value tolerance is achieved.

Parameters

func	Input lambda function to be solved for \(f(x)=0\), takes real value and returns real value
lower_bound	lower bound of the search, \(a\)
upper_bound	upper bound of the search, \(b\)
rel_tolerance	Relative tolerance scale, stops search when \(\left|f(x_i)\right|<\epsilon \left|f(a)-f(b)\right|\)
abs_tolerance	Absolute tolerance scale, stops search when \(\left|f(x_i)\right|<\epsilon\)

Definition at line 619 of file size_field.cpp.

evaluate_complexity()

template<int dim, typename real >

real PHiLiP::GridRefinement::SizeField< dim, real >::evaluate_complexity ( const dealii::DoFHandler< dim > &  dof_handler, const dealii::hp::MappingCollection< dim > &  mapping_collection, const dealii::hp::FECollection< dim > &  fe_collection, const dealii::hp::QCollection< dim > &  quadrature_collection, const dealii::UpdateFlags &  update_flags, const std::unique_ptr< Field< dim, real >> &  h_field, const dealii::Vector< real > &  p_field  )

staticprotected

Evaluates the continuous complexity (approximate degrees of freedom) for the target \(hp\) mesh representation.

Discrete complexity approximation is obtained by considering the frame-field target distribution (with either isotropic or anisotropic field distribution) and the vector of \(C^0\) polynomial distribution. Together these fully describe the target \(hp\) mesh and in the continuous space the degrees of freedom can be approximated by:

\[ \mathcal{C}(\mathcal{M}) = \int_{\Omega} { \mathrm{det}{(V(\boldsymbol{x}))} \left(\mathcal{P}(\boldsymbol{x})+1\right)^2 \mathrm{d}\boldsymbol{x}} \]

or by a summation based on the elements of the discrete existing mesh:

\[ \mathcal{C} = \sum_{i=0}^{N} { \mathrm{det}{(V_i^{-1})} \left(p_i+1\right)^2 } \]

This is commonly used as a target parameter for controlling leve of refinement in conjunction with a bisection function as used in many of the methods listed above.

Parameters

dof_handler	DoFHandler describing the mesh
mapping_collection	Element mapping collection
fe_collection	Finite element collection
quadrature_collection	Quadrature rules collection
update_flags	Update flags for the volume finite elements
h_field	(Input) Current size-field
p_field	(Input) Current polynomial field

Definition at line 99 of file size_field.cpp.

isotropic_h()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::isotropic_h ( const real  complexity, const dealii::Vector< real > &  B, const dealii::DoFHandler< dim > &  dof_handler, const dealii::hp::MappingCollection< dim > &  mapping_collection, const dealii::hp::FECollection< dim > &  fe_collection, const dealii::hp::QCollection< dim > &  quadrature_collection, const dealii::UpdateFlags &  update_flags, std::unique_ptr< Field< dim, real >> &  h_field, const dealii::Vector< real > &  p_field  )

static

Computes the size field (element scale) for a potentially non-uniform \(p\)- distribution.

Based on the same idea as isotropic_uniform size field method. The continuous constained error minimization problem can again be expressed based on \(B(x)\), the error distribution from the quadratic form of \(p+1\) directional derivatives:

\[ \min_{d} \quad \mathcal{E}(d,\mathcal{P}) = \int_{\Omega} { B(\boldsymbol{x},\mathcal{P}(\boldsymbol{x})) d(\boldsymbol{x})^{-\frac{q(\mathcal{P}(\boldsymbol{x})+1)}{2}} \mathrm{d}\boldsymbol{x}} \]

\[ \textrm{s.t.} \quad \mathcal{N}_{hp}(d, \mathcal{P}) = \int_{\Omega} { d(\boldsymbol{x}) (\mathcal{P}(\boldsymbol{x}))+1)^2 \mathrm{d}\boldsymbol{x}} \leq \mathcal{C}_{t} \]

However, at this point the global problem cannot be solved analytically due to the inclusion of the polynomial distribution \(\mathcal{P}(\boldsymbol{x})\). Instead, it can be rewritten as a lagragian which leads to

\[ h(\boldsymbol{x})^{dim} = \left( \frac{2\lambda(\mathcal{P}(\boldsymbol{x})+1)} {q B(\boldsymbol{x}, \mathcal{P}(\boldsymbol{x}))} \right)^{\frac{-2}{-q(\mathcal{P}(\boldsymbol{x})+1)+2}} \]

The overall problem is then solved by performing bisection on this \(\lambda\) parameter until the complexity target is suitably matched. Thus allowing the updated size distribution to be directly extracted from this expression.

Parameters

complexity	Target global continuous mesh complexity
B	Continuous error model for \(p+1\) directional derivatives
dof_handler	DoFHandler describing the mesh
mapping_collection	Element mapping collection
fe_collection	Finite element collection
quadrature_collection	Quadrature rules collection
update_flags	Update flags for the volume finite elements
h_field	(Output) Target size-field
p_field	(Input) Current polynomial field

Definition at line 70 of file size_field.cpp.

isotropic_hp()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::isotropic_hp ( const real  complexity, const dealii::Vector< real > &  Bm, const dealii::Vector< real > &  B, const dealii::Vector< real > &  Bp, const dealii::DoFHandler< dim > &  dof_handler, const dealii::hp::MappingCollection< dim > &  mapping_collection, const dealii::hp::FECollection< dim > &  fe_collection, const dealii::hp::QCollection< dim > &  quadrature_collection, const dealii::UpdateFlags &  update_flags, std::unique_ptr< Field< dim, real >> &  h_field, dealii::Vector< real > &  p_field  )

static

Computes the size field (element scale) and updated polynomial distrubition for high-order error function.

Based on the method of Dolejsi et al. originally developped for the simplex case. See: Dolejsi et al. 2018 "A continuous hp-mesh model for adaptive discontinuous Galerkin schemes"

Overall, the process involves solving for an optimal \(h\)- distribution using the method outlined in isotropic_h above followed by a sampling of changes to the local polynomial orders. Here, additional error distributions for the \(p+1\) quadratic derivative forms must be provided for an approximation of a coarsened (error is current largest terms) or refined polynomial order (error is two orders higher). After convergence of the size fields, this amounts to minimizing the error at a constant complexity level based on the error distribution integrand:

\[ e_{ref}(\boldsymbol{x}) = \left( B_{p}(\boldsymbol{x}) h(\boldsymbol{x})^{\frac{dim(p+1)}{2}} \right)^q \]

\[ N_{ref}(\boldsymbol{x}) = \left( \frac{p+1}{h(\boldsymbol{x})} \right)^{dim} \]

Based on the equal complexity \(N_{ref} = N_{p-1} = N_{p-1}\) the equivalent local sizes are:

\[ h_{p-1}(\boldsymbol{x}) = \frac{p} {N_{ref}^{\frac{1}{dim}}} \]

\[ h_{p+1}(\boldsymbol{x}) = \frac{p+2} {N_{ref}^{\frac{1}{dim}}} \]

Where the lower and higher polynomial distributions lead to error estimates of:

\[ e_{p-1}(\boldsymbol{x}) = \left( B_{p-1}(\boldsymbol{x}) h_{p-1}(\boldsymbol{x})^{\frac{dim(p)}{2}} \right)^q \]

\[ e_{p+1}(\boldsymbol{x}) = \left( B_{p+1}(\boldsymbol{x}) h_{p+1}(\boldsymbol{x})^{\frac{dim(p+2)}{2}} \right)^q \]

Where the polynomial with lowest error is then chosen for the \(p\)- field. Note: not thoroughly tested due to limitations in the \(p\)- adaptivity.

Parameters

complexity	Target global continuous mesh complexity
Bm	Continuous error model for \(p\) directional derivatives
B	Continuous error model for \(p+1\) directional derivatives
Bp	Continuous error model for \(p+2\) directional derivatives
dof_handler	DoFHandler describing the mesh
mapping_collection	Element mapping collection
fe_collection	Finite element collection
quadrature_collection	Quadrature rules collection
update_flags	Update flags for the volume finite elements
h_field	(Output) Target size-field
p_field	(Output) Target polynomial field

Definition at line 206 of file size_field.cpp.

isotropic_uniform()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::isotropic_uniform ( const real &  complexity, const dealii::Vector< real > &  B, const dealii::DoFHandler< dim > &  dof_handler, std::unique_ptr< Field< dim, real >> &  h_field, const real &  poly_degree  )

static

Computes the size field (element scale) for a uniform \(p\)- distribution.

Unlike other cases, here for a constant value of \(p\)-, the contiunuous constrained error minimization problem can be written by the set of integrals involving the mesh density distribution \(d(x)\)

\[ \min_{d} \quad \mathcal{E}(d,\mathcal{P}) = \int_{\Omega} { B(\boldsymbol{x},\mathcal{P}(\boldsymbol{x})) d(\boldsymbol{x})^{-\frac{q(\mathcal{P}(\boldsymbol{x})+1)}{2}} \mathrm{d}\boldsymbol{x}} \]

\[ \textrm{s.t.} \quad \mathcal{N}_{hp}(d, \mathcal{P}) = \int_{\Omega} { d(\boldsymbol{x}) (\mathcal{P}(\boldsymbol{x}))+1)^2 \mathrm{d}\boldsymbol{x}} \leq \mathcal{C}_{t} \]

where \(B(x)\) is the error distribution based on the approximate quadratic form obtained by reconstruction of the \(p+1\) directional derivatives. This can be solved directly by calculus of variations leading to the global distribution for the scaling as:

\[ h(\boldsymbol{x})^{dim} = \left( \frac {C_t B(\boldsymbol{x})} {\int_{\Omega}{ B(\boldsymbol{x})^{\frac{2}{q(p+1)+2}} \mathrm{d}\boldsymbol{x}}} \right)^{\frac{-2}{q(p+1)+2}} \]

The main difference here is that the \(\lambda\) bisection parameter can be plugged back through the complexity constraint and removed the integral giving the simplified final form.

Parameters

complexity	Target global continuous mesh complexity
B	Continuous error model for \(p+1\) directional derivatives
dof_handler	DoFHandler describing the mesh
h_field	(Output) Target size-field
poly_degree	(Input) Uniform polynomial degree

Definition at line 38 of file size_field.cpp.

update_alpha_k_balan()

template<int dim, typename real >

real PHiLiP::GridRefinement::SizeField< dim, real >::update_alpha_k_balan ( const real  eta_k, const real  r_max, const real  c_max, const real  eta_min, const real  eta_max, const real  eta_ref  )

staticprotected

Determines local \(\alpha\) sizing factor (from adjoint estimates)

Uses relative scale of local Dual-Weighted Residual (DWR) relative to global maximum and minimums and a chosen threshold value based on target output complexity. Sizing update is basaed on Eq. 30-33 from Balan et al. "Adjoint-based hp-adaptivity on anisotropic mesh for high-order..." where the scaling is fit quadratically to a predefined range in the logarithmic space:

\[ \alpha_{k}=\left\{\begin{array}{ll} \left(\left(r_{max }-1\right) \xi_{k}^{2}+1\right)^{-1}, & \eta_{k} \geq \eta_{ref}, \\ \left(\left(c_{max}-1\right) \xi_{k}^{2}+1\right), & \eta_{k}<\eta_{ref}, \end{array}\right. \]

where

\[ \xi_{k}= \left\{\begin{array}{ll} \frac{\log \left(\eta_{k}\right)-\log \left(\eta_{ref}\right)}{\log \left(\eta_{max }\right)-\log \left(\eta_{ref}\right)}, & \eta_{k} \geq \eta_{ref}, \\ \frac{\log \left(\eta_{k}\right)-\log \left(\eta_{ref}\right)}{\log \left(\eta_{min }\right)-\log \left(\eta_{ref}\right)}, & \eta_{k}<\eta_{ref}. \end{array}\right. \]

This function returns the value \(\alpha\) used to update the local size measure.

Parameters

eta_k	Local value of DWR indicator
r_max	Maximum refinement scaling factor
c_max	Maximum coarsening scaling factor
eta_min	Minimum value of DWR indicator
eta_max	Maximum value of DWR indicator
eta_ref	Threshold value of DWR for deciding between coarsening and refinement

Definition at line 577 of file size_field.cpp.

update_alpha_vector_balan()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::update_alpha_vector_balan ( const dealii::Vector< real > &  eta, const real  r_max, const real  c_max, const real  eta_min, const real  eta_max, const real  eta_ref, const dealii::DoFHandler< dim > &  dof_handler, const dealii::Vector< real > &  I_c, std::unique_ptr< Field< dim, real >> &  h_field  )

staticprotected

Updates the size field for the mesh through the area measure based on reference value.

Updates the h-field object based on an input vector or error values and reference parameters. Mainly called as the update for bisection from adjoint_h_balan. Based on the relative local error determines a scaling factor from update_alpha_balan, scales the local h field according to the growth in cell measure \(I_k\) (length in 1D, area in 2D, volume in 3D):

\[ I_k = \alpha_k I_k^c \]

where \(I_k^c\) is the initial coarse cell size. In the 2D quad case for the frame field defintion \ this is equivalent to updating the local scaling factors by:

\[ h = \sqrt{\frac{1}{4} \alpha_k I_k^c} \]

Overall, this shifts the problem of complexity targetting to sampling different values of \(\eta_{ref}\).

Parameters

eta	Dual-weighted residual (DWR) error indicator distribution
r_max	Maximum refinement scaling factor
c_max	Maximum coarsening scaling factor
eta_min	Minimum value of DWR indicator
eta_max	Maximum value of DWR indicator
eta_ref	Threshold value of DWR for deciding between coarsening and refinement
dof_handler	DoFHandler describing the mesh
I_c	Vector of cell current cell area measure
h_field	(Output) Updated size-field

Definition at line 535 of file size_field.cpp.

update_h_dwr()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::update_h_dwr ( const real  tau, const dealii::Vector< real > &  eta, const dealii::DoFHandler< dim > &  dof_handler, std::unique_ptr< Field< dim, real >> &  h_field, const real &  poly_degree  )

staticprotected

Updates the size field based on the DWR disitribution and a reference value, \(\tau\).

This assumes that the error in the DWR will converge at the optimal \(2p+1\) rate and attempts to equally distribute this error over the mesh complexity. Locally, with suitable choice of this parameter this leads to the size field distribution from:

\[ h(\boldsymbol{x}) = \left( \frac{\tau}{\eta(\boldsymbol{x})} \right)^{\frac{1}{2p+1}} \]

Note: This method is not recommended as it leads to large oscillations in the size field because the error does not predictibly follow this convergence pattern.

Parameters

tau	Bisection reference value for scaling
eta	Dual-weighted residual (DWR) error indicator distribution
dof_handler	DoFHandler describing the mesh
h_field	(Output) Target size-field
poly_degree	(Input) Uniform polynomial degree

Definition at line 507 of file size_field.cpp.

update_h_optimal()

template<int dim, typename real >

void PHiLiP::GridRefinement::SizeField< dim, real >::update_h_optimal ( const real  lambda, const dealii::Vector< real > &  B, const dealii::DoFHandler< dim > &  dof_handler, std::unique_ptr< Field< dim, real >> &  h_field, const dealii::Vector< real > &  p_field  )

staticprotected

Based on a chosen bisection paraemeter, updates the \(h\)- field to an optimal distribution.

This is based on the solution to the error minimization problem. Overall, given some choice of paratere \(\lambda\), the size field should take the form of:

\[ d(\boldsymbol{x}) = \left( \frac{2\lambda(\mathcal{P}(\boldsymbol{x})+1)} {q B(\boldsymbol{x}, \mathcal{P}(\boldsymbol{x}))} \right)^{\frac{2}{-q(\mathcal{P}(\boldsymbol{x})+1)+2}} \]

where \(d(\boldsymbol{x})\) is the local mesh density ( \(\approx h(\boldsymbol{x})^{-2}\) for 2D quad meshing) and \(B(\boldsymbol{x})\) is the quadratic error distribution. The overall process can be controlled by checking the resulting continuous complexity of the result until the error problem constraint is suitably satisfied.

Parameters

lambda	(Input) Current complexity bisection parameter
B	Continuous error constant for quadratic model of \(p+1\) directional derivatives
dof_handler	DoFHandler describing the mesh
h_field	(Input) Target size-field
p_field	(Input) Current polynomial field

Definition at line 143 of file size_field.cpp.

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The documentation for this class was generated from the following files:

• src/grid_refinement/size_field.h
• src/grid_refinement/size_field.cpp