Muscat.MeshTools.AnisotropicMetricComputation module

BuildMet(eigenValue: ndarray[tuple[Any, ...], dtype[_ScalarT]], eigenVector: ndarray[tuple[Any, ...], dtype[_ScalarT]]) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Build metric from eigenValue and eigenVector

Parameters:

  • eigenValue – NDArray computed eigenValues for metric on nodes

  • eigenVector – NDArray eigenVectors for metric on nodes

Returns:

NDArray

a vector containing the (dim+1)*dim/2 coefficients of the associated symmetric definite positive matrix

Return type:

metric

CheckIntegrity(GUI: bool = False) → str[source]
CheckIntegrity_ComputeMetric(GUI=False)[source]
CheckIntegrity_GeoMeanOfMatrices(GUI=False)[source]
CheckIntegrity_IntersectionOfMetrics(GUI=False)[source]
ComputeDensity(mesh: Mesh, eigenValue: ndarray[tuple[Any, ...], dtype[_ScalarT]], ratio: ndarray[tuple[Any, ...], dtype[_ScalarT]], err: float64 = None, N: int64 = 1, p: int64 = None, renormalization: bool = True) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Construct at each node of the mesh the density d = 1 /(h_1 h_2 h_3) where h_i is the length of the optimal metric

It is given by d = N VO det / integralOfdet where det = (sum_i ratio_i^(2/dim) * gamma_i )^(dim*p/(2*p+dim)))

Parameters:

  • mesh (Mesh) – the input mesh

  • eigenValue (NDArray) – the absolute values of the eigenValues of the hessian

  • err (MuscatFloat) – desired level for the absolute interpolation error

  • N (MuscatIndex) – desired number of elements in the adapted mesh

  • p (MuscatIndex) – order of the Lp-norm (infinite by default)

Returns:

density

Return type:

MuscatFloat

ComputeGradient(mesh: Mesh, psi: ndarray[tuple[Any, ...], dtype[_ScalarT]], tags: List[str] = None) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Compute the gradient at each node from a scalar nodefield using a variational formulation: Find Grad such that int Grad . GradT dx = int grad(Psi) . GradT dx for all GradT

Parameters:

  • mesh (Mesh) – the input mesh

  • psi (NDArray) – a scalar nodefield

  • tags (List[str]) – eTags of the computational domain, if None the domain is the entire mesh

Returns:

grad – the gradient of psi at each node

Return type:

NDarray

ComputeGradient_L2projection(mesh: Mesh, psi: ndarray[tuple[Any, ...], dtype[_ScalarT]], tags: List[str] = None) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Compute the gradient at each node from a scalar nodefield psi using formula: grad_i(psi) = sum_{K in S_i} |K| grad_K(psi) / sum_{K in S_i} |K|

Warning : works only with simplex linear meshes

Parameters:

  • mesh (Mesh) – the input mesh

  • psi (NDArray) – a scalar nodefield

  • tags (List[str]) – eTags of the computational domain, if None the domain is the entire mesh

Returns:

grad – the gradient of psi at each node

Return type:

NDarray

ComputeHessian(mesh: Mesh, psi: ndarray[tuple[Any, ...], dtype[_ScalarT]], gradL2: bool = False, tags: List[str] = None) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Compute the hessian at each node from a scalar nodefield phi using twice the ComputeGradient function

Parameters:

  • mesh (Mesh) – the input mesh

  • psi (NDArray) – a scalar nodefield

  • tags (List[str]) – eTags of the computational domain, if None the domain is the entire mesh

  • gradL2 (bool) – if True, use a L2 projection to compute the gradients

Returns:

hess – the symmetric matrix hessian of psi at each node

Return type:

NDarray

ComputeMetric(mesh: Mesh, psi: ndarray[tuple[Any, ...], dtype[_ScalarT]], err: float64 = None, N: int64 = 1, p: int64 = None, gradL2: bool = False, zones: List[List[str]] = None, isotropic: bool = False) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Construct a metric field for mesh adaptation from a given scalar nodefield phi in Lp norm From article Continuous mesh framework part II: validations and applications by Adrien Loseille and Frédéric Alauzet, SIAM Journal of Numerical Analysis, 2011.

Parameters:

  • mesh (Mesh) – the input mesh

  • psi (NDArray) – a scalar nodefield

  • p (MuscatIndex) – order of the Lp-norm (infinite by default)

  • err (MuscatFloat) – desired level for the relative interpolation error (with respect to normLp(psi))

  • N (MuscatIndex) – desired number of elements in the adapted mesh

  • zones (List[List[str]]) – list of the eTags of the computational subdomains, if None, the entire domain is considered

  • isotropic (bool, default False) – if True, an isotropic scalar metric is returned

Returns:

metric – the metric at each node a metric is either a vector of the sizes of the cells (if option isotropic = True) or a vector containing the (dim+1)*dim/2 coefficients of the associated symmetric definite positive matrix warning : order of metric component imposed by mmg3d, different from order read by paraview

Return type:

NDarray

ComputeMetricFieldOfExistingMesh(mesh: Mesh)[source]

Compute a metric field corresponding to the input existing mesh

Parameter

meshMesh

the input mesh

returns:

metP1 – the metric field at each node

rtype:

NDarray

ComputeMetricOnSubdomain(mesh: Mesh, psi: ndarray[tuple[Any, ...], dtype[_ScalarT]], err: float64 = None, N: int64 = 1, p: int64 = None, gradL2: bool = False, tags: List[str] = None, isotropic: bool = False, renormalization: bool = True) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Construct a metric field for mesh adaptation from a given scalar nodefield phi in Lp norm From article Continuous mesh framework part II: validations and applications by Adrien Loseille and Frédéric Alauzet, SIAM Journal of Numerical Analysis, 2011.

Parameters:

  • mesh (Mesh) – the input mesh

  • psi (NDArray) – a scalar nodefield

  • p (MuscatIndex) – order of the Lp-norm (infinite by default)

  • err (MuscatFloat) – desired level for the absolute interpolation error

  • N (MuscatIndex) – desired number of elements in the adapted mesh

  • tags (List[str]) – eTags of the computational subdomain, if None, the entire domain is considered

  • isotropic (bool, default False) – if True, an isotropic scalar metric is returned

  • flag (bool, default True) – if False, the global density is set to 1

Returns:

metric – the metric at each node a metric is either a vector of the sizes of the cells (if option isotropic = True) or a vector containing the (dim+1)*dim/2 coefficients of the associated symmetric definite positive matrix warning : order of metric component imposed by mmg3d and Inria Tools, different from order read by paraview

Return type:

NDarray

ComputeRatio(eigenValue: ndarray[tuple[Any, ...], dtype[_ScalarT]]) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Construct at each node of the mesh the ratios r_i = h_i^3 /(h_1 h_2 h_3) where h_i is the length of the optimal metric It is given by r_1 = gamma_1^-1 * gamma_2^1/2 * gamma_3^1/2

Parameters:

eigenValue (NDArray) – the absolute values of the eigenValues of the hessian

Returns:

ratio

Return type:

NDarray

GeometricMeanOfMatrices(metricIn: ndarray[tuple[Any, ...], dtype[_ScalarT]], weightAtElement: ndarray[tuple[Any, ...], dtype[_ScalarT]] = None, conn: ndarray[tuple[Any, ...], dtype[_ScalarT]] = None, n_nodes: int64 = None) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

This function computes the log-euclidian mean of a bunch of matrices. It is a good approximation of the Karcher mean described in A SURVEY AND COMPARISON OF CONTEMPORARY ALGORITHMS FOR COMPUTING THE MATRIX GEOMETRIC MEAN, BEN JEURIS, RAF VANDEBRIL AND BART VANDEREYCKEN, 2012.

Parameters:

  • matrices (NDarray) – a bunch of symmetric definite positive matrices of the same dimension

  • weights (NDarray) – the corresponding weights for the mean, we renormalize the sum of the weights to 1

Returns:

metricOut – the weighted log-euclidian mean

Return type:

NDarray

IntersectionOfMetrics(metrics: List, active_scaling: bool = True, preprocessing: bool = True) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Construct the Löwner-John metric from a set of metrics From article LMI approximations for the radius of the intersection of ellipsoids: a survey, D. Henrion, S. Tarbouriech, D. Arzelier, 1998

Parameters:

  • metrics (List of NDarrays) – a list of metric nodefields of the same dimension a metric is a vector containing the dimF = dim*(dim+1)/2 coefficients of the symmetric definite positive matrix

  • active_preprocessing (bool) –

    Activate or not the pre processing of the metrics to improve clarabel convergence active_scaling : bool

    Activate or not the scaling of the metrics

Returns:

metric – at each node, the Löwner-John metric corresponding to the largest ellipsoid included in the intersection of the input metrics

Return type:

NDarray

MetricFromElementsToNodes(mesh: Mesh, metricIn: ndarray[tuple[Any, ...], dtype[_ScalarT]]) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]

Compute the metric field defined at the nodes, corresponding to the geometric mean of a metric field defined at the elements Warning : currently only works with linear simplex mesh

Parameters:

  • mesh (Mesh) – the input mesh

  • metricsIn (NDarrays) – a metric field defined at each element of the mesh a metric is a vector containing the dimF = dim*(dim+1)/2 coefficients of the symmetric definite positive matrix

Returns:

metricOut – a metric field defined at each node of the mesh

Return type:

NDarrays

ObviousSolutionOrScale(node_metrics: ndarray[tuple[Any, ...], dtype[_ScalarT]], active_scaling: bool = True) → Tuple[bool, int64, float64][source]

Look if one of the metric to intersect at node is obviously solution, otherwise compute scaling to apply to help solver CLARABEL to compute intersection problem

Parameter

node_metricsNDarrays

an array of metrics of the same shape at a given node a metric is a vector containing the dimF = dim*(dim+1)/2 coefficients of the symmetric definite positive matrix

active_scalingbool

Activate or not the scaling of the metrics to improve clarabel convergence

returns:

  • obvious (bool) – Is there a metric obviously solution

  • index (int) – Index of the metric solution

  • scale (float) – Scaling factor to apply to the metrics to help solver CLARABEL converge

PlotMetric(list_met: List)[source]

Function to generate a matplolib figure of ellipses/ellipsoids representing metrics

Parameters:

  • list_met (List) – List of metrics to plot as ellipses,

  • format (metrics written under flat mmg)

Returns:

figure that can be plot or saved

Return type:

fig (Matplotlib figure)

expm_sym(S: ndarray[tuple[Any, ...], dtype[_ScalarT]]) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]
logm_spd(A: ndarray[tuple[Any, ...], dtype[_ScalarT]], eps: float = 1e-15) → ndarray[tuple[Any, ...], dtype[_ScalarT]][source]