meshQuality

Evaluate shape quality of mesh elements

Syntax

Q = meshQuality(mesh)

Q = meshQuality(mesh,elemIDs)

Q = meshQuality(___,"aspect-ratio")

Description

Q = meshQuality(mesh) returns a row vector of numbers from 0 through 1 representing shape quality of all elements of the mesh. Here, 1 corresponds to the optimal shape of the element.

Q = meshQuality(mesh,elemIDs) returns the shape quality of the specified elements.

Q = meshQuality(___,"aspect-ratio") determines the shape quality by using the ratio of minimal to maximal dimensions of an element. The quality values are numbers from 0 through 1, where 1 corresponds to the optimal shape of the element. Specify "aspect-ratio" after any of the previous syntaxes.

Examples

Element Quality of 3-D Mesh

Evaluate the shape quality of the elements of a 3-D mesh.

Create a PDE model.

model = createpde;

Include and plot the following geometry.

importGeometry(model,"PlateSquareHoleSolid.stl");
pdegplot(model)

Create and plot a coarse mesh.

mesh = generateMesh(model,"Hmax",35)

mesh = 
  FEMesh with properties:

             Nodes: [3x838 double]
          Elements: [10x364 double]
    MaxElementSize: 35
    MinElementSize: 17.5000
     MeshGradation: 1.5000
    GeometricOrder: 'quadratic'

pdemesh(model)

Evaluate the shape quality of all mesh elements. Display the first five values.

Q = meshQuality(mesh);
Q(1:5)

ans = 1×5

    0.8692    0.7119    0.8047    0.7838    0.8417

Find the elements with the quality values less than 0.2.

elemIDs = find(Q < 0.2);

Highlight these elements in blue on the mesh plot.

pdemesh(mesh,"FaceAlpha",0.5)
hold on
pdemesh(mesh.Nodes,mesh.Elements(:,elemIDs), ...
                  "FaceColor","blue", ...
                  "EdgeColor","blue")

Plot the element quality in a histogram.

figure
hist(Q)
xlabel("Element Shape Quality","fontweight","b")
ylabel("Number of Elements","fontweight","b")

Find the worst quality value.

Qworst = min(Q)

Qworst = 
0.1903

Find the corresponding element IDs.

elemIDs = find(Q==Qworst)

elemIDs = 1×2

     8   244

Element Quality of 2-D Mesh

Evaluate the shape quality of the elements of a 2-D mesh.

Create a PDE model.

model = createpde;

Include and plot the following geometry.

importGeometry(model,"PlateSquareHolePlanar.stl");
pdegplot(model)

Create and plot a coarse mesh.

mesh = generateMesh(model,"Hmax",20)

mesh = 
  FEMesh with properties:

             Nodes: [2x298 double]
          Elements: [6x132 double]
    MaxElementSize: 20
    MinElementSize: 10
     MeshGradation: 1.5000
    GeometricOrder: 'quadratic'

pdemesh(model)

Find the IDs of the elements within a box enclosing the center of the plate.

elemIDs = findElements(mesh,"box",[25,75],[80,120]);

Evaluate the shape quality of these elements. Display the result as a column vector.

Q = meshQuality(mesh,elemIDs);
Q.'

ans = 20×1

    0.8158
    0.4618
    0.9839
    0.5948
    0.3790
    0.3957
    0.3599
    0.6534
    0.9919
    0.6130
      ⋮

Find the elements with the quality values less than 0.4.

elemIDs04 = elemIDs(Q < 0.4)

elemIDs04 = 1×3

    45    51    57

Highlight these elements in green on the mesh plot. Zoom in to see the details.

pdemesh(mesh,"ElementLabels","on")
hold on
pdemesh(mesh.Nodes,mesh.Elements(:,elemIDs04),"EdgeColor","green")
zoom(10)

Element Quality Determined by Aspect Ratio

Determine the shape quality of mesh elements by using the ratios of minimal to maximal dimensions.

Create a PDE model and include the L-shaped geometry.

model = createpde(1);
geometryFromEdges(model,@lshapeg);

Generate the default mesh for the geometry.

mesh = generateMesh(model);

View the mesh.

pdeplot(model)

Evaluate the shape quality of mesh elements by using the minimal to maximal dimensions ratio. Display the first five values.

Q = meshQuality(mesh,"aspect-ratio");
Q(1:5)

ans = 1×5

    0.9964    0.9964    0.5385    0.9605    0.9566

Evaluate the shape quality of mesh elements by using the default setting. Display the first five values.

Q = meshQuality(mesh);
Q(1:5)

ans = 1×5

    1.0000    1.0000    0.8261    0.9992    0.9990

Input Arguments

mesh — Mesh object

Mesh property of a PDEModel object | output of generateMesh

Mesh object, specified as the Mesh property of a PDEModel object or as the output of generateMesh.

Example: model.Mesh

elemIDs — Element IDs

positive integer | matrix of positive integers

Element IDs, specified as a positive integer or a matrix of positive integers.

Example: [10 68 81 97 113 130 136 164]

Output Arguments

Q — Shape quality of mesh elements

row vector of numbers from 0 through 1

Shape quality of mesh elements, returned as a row vector of numbers from 0 through 1. The value 0 corresponds to a deflated element with zero area or volume. The value 1 corresponds to an element of optimal shape.

Example: [0.9150 0.7787 0.9417 0.2744 0.9843 0.9181]

Data Types: double

Algorithms

By default, meshQuality calculates the shape quality of a triangular mesh element as follows:

$$Q=\frac{4\sqrt{3}A}{{\displaystyle \sum _{i=1}^3{l}_i^2}}$$

Here, A is the area of the triangle, and l_i are the edge lengths of the triangle.

meshQuality calculates the shape quality of a tetrahedral mesh element as follows:

$$Q=\frac{18V}{\sqrt{{\displaystyle \sum _{i=1}^6{l}_i^2}}\sqrt{{\displaystyle \sum _{k=1}^4{A}_k^2}}}$$

Here, V is the volume of the tetrahedron, l_i are the edge lengths, and A_k are the areas of the triangular faces.

When you use the aspect-ratio argument, meshQuality calculates the quality of a triangular mesh element as follows:

$$Q_{AR}=\frac{4}{\sqrt{3}}\frac{A}{l_{\max }^2}$$

Here, l_max is the maximal edge length of the triangle.

The aspect-ratio quality of a tetrahedral mesh element is:

$$Q_{AR}=\frac{\text{3}\sqrt{6}}{2}\frac{\text{V}}{{\text{l}}_{\text{max}}{A}_{\text{max}}}$$

Here, l_max is the maximal edge length of the tetrahedron, and A_max is the area of the largest face of the tetrahedron:

$$A_{\max }=\max (A_k),k=1,\dots ,4$$

References

[1] Knupp, Patrick M. "Matrix Norms & the Condition Number: A General Framework to Improve Mesh Quality via Node-Movement." In Proceedings, 8th International Meshing Roundtable. Lake Tahoe, CA, October 1999: 13-22.

[2] Shewchuk, Jonathan R. "What Is a Good Linear Element? Interpolation, Conditioning, and Quality Measures." In Proceedings, 11th International Meshing Roundtable. Ithaca, NY, September 2002: 115-126.

Version History

Introduced in R2018a