On the accuracy of dG discretizations on curved and agglomerated - - PowerPoint PPT Presentation

Workshop on Polygonal and Polyhedral Meshes, Università di Milano Bicocca, 17-19 settembre 2012.

On the accuracy of dG discretizations

• n curved and agglomerated elements meshes

Lorenzo Botti , Francesco Bassi , Alessandro Colombo . Università degli Studi di Bergamo, Dipartimento di Ingegneria Industriale.

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Outline

• 1. Compare physical and reference frame dG discretizations on

curved elements meshes

• 2. Evaluate the accuracy on agglomerated elements obtained

clustering together the cells of a standard fine grid

• 3. Agglomeration based hemodynamic application

Exploit the flexibility of agglomeration based dG discretizations in complex 3D domains

• L2-orthogonal projection (approximation properties).
• INS dG discretization, Kovasznay flow (efficiency).

Motivation

dG methods are well suited in the context of conservation laws

• Conservation of physical quantities
• Robustness in convection dominated flows
• Accuracy near boundaries to resolve boundary layers

dG discretizations are expensive

Exploit the geometrical flexibility provided by agglomeration to avoid high-order mesh generation techniques.

Why dG methods on general meshes? Curved elements meshes are commonly employed to High-order methods need accurate domain approximations Challenges we need to face...

To justify the computational effort the accuracy should be maintained also on curved element meshes.

• Increase the accuracy of the domain discretization
• Decrease the number of elements to perform higher-order discretizations

Accuracy on general meshes

• the element shape
• the mapping function
• the discrete polynomial space

The discrete space choice might have a significant impact on the accuracy. h-convergence rates associated to reference frame discretization might degrade depending on Consider a mesh composed of quadrilateral elements .

Discrete polynomial space choice

• Physical frame discretizations (PFD)
• Reference frame discretizations (RFD)

where is a polynomial mapping such that and is the reference square. see Arnold, Boffi, Falk [Math. Comp., 2002].

Accuracy on general meshes, physical frame

Physical frame discretizations provide optimal approximation properties

• n regular mesh sequences, that is

where is the -orthogonal projection onto and the constant depends only on a mesh regularity parameter and the polynomial degree k. Optimal approximation properties for regular mesh sequences can be proved using the version of the Bramble-Hilbert Lemma introduced by Brenner and Scott [The mathematical theory of finite element methods, Springer 2008], based on averaged Taylor polynomials. See also [Botti, J. Sci. Comput., 2012]. Mesh regularity implies that each mesh element E is star-shaped with respect to a ball with uniformly comparable diameter with respect to .

Accuracy on general meshes, physical frame

Physical frame discretizations provide optimal approximation properties

• n regular mesh sequences, that is

where is the -orthogonal projection onto and the constant depends only on a mesh regularity parameter and the polynomial degree k. Mesh regularity implies that each mesh element E is star-shaped with respect to a ball with uniformly comparable diameter with respect to .

Accuracy on general meshes, reference frame

We define the effective mapping order associated to the polynomial space as the minimum positive integer such that

, where .

Examples

curved 9-node quadrilateral, mapping space curved 8-node quadrilateral, mapping space

Accuracy on general meshes, reference frame

We define the effective mapping order associated to the polynomial space as the minimum positive integer such that

, where .

Examples

curved 9-node quadrilateral, mapping space 8-node rectangular element, effective mapping space

Accuracy on general meshes, reference frame

The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection

• perator onto the space , where .

We define the effective mapping order associated to the polynomial space as the minimum positive integer such that

, where .

Accuracy on general meshes, reference frame

The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection

• perator onto the space , where .

We define the effective mapping order associated to the polynomial space as the minimum positive integer such that

, where .

We consider the approximation of a function v over an element E using the space , the error in norm can be computed as

Sketch of the proof

Accuracy on general meshes, reference frame

Since, as suggested by Arnold, Boffi, Falk [Math. Comp., 2002] We define the effective mapping order associated to the polynomial space as the minimum positive integer such that

, where .

we are able to infer that The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection

• perator onto the space , where .

Accuracy on general meshes, reference frame

see Botti [J. Sci. Comput., 2012] for details. We define the effective mapping order associated to the polynomial space as the minimum positive integer such that

, where .

Using the effective mapping order definition we get The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection

• perator onto the space , where .

Accuracy on general meshes

Distorted second-order quadrilateral meshes of the unit square Second-order quadrilateral meshes approximating the unit circle Compare the accuracy of physical and reference frame discretizations. Consider the L2 projection of y^6 on regular mesh sequences

Accuracy on general meshes, square mesh sequence

Reference frame discretizations Physical frame discretizations

10 100 1000

sqrt(number of elements)

1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1

error in L2 norm

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 k=11 k=12 k=13 k=14 k=15 nine-node quadrilaterals

Convergence rates (error in L2 norm)

• Reference frame discretizations
• Physical frame discretizations

10 100 1000

sqrt(number of elements)

1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1

error in L2 norm

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 k=11 eight-node quadrilaterals 10 100 1000

sqrt(number of elements)

1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1 1

error in L2 norm

k=0 k=1 k=2 k=3 k=4 k=5 eight and nine-node quadrilaterals

Accuracy on general meshes, circular mesh sequence.

Reference frame discretizations Physical frame discretizations

Since for most of the mesh elements , the convergence rates of reference frame discretizations are not spoiled.

10 100

sqrt(number of elements)

1x10-14 1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1

error in L2 norm

k=0 k=1 k=2 k=3 k=4 k=5 10 100

sqrt(number of elements)

1x10-14 1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1

error in L2 norm

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8

Accuracy on general meshes, Circular mesh family.

Reference frame discretizations Physical frame discretizations

Since for most of the mesh elements , the convergence rates of reference frame discretizations are not spoiled. Physical frame discretizations provide the best accuracy for a given number of dofs.

10 100

sqrt(number of elements)

1x10-14 1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1

error in L2 norm

k=6 k=0 k=1 k=2 k=3 k=4 k=5 10 100

sqrt(number of elements)

1x10-14 1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 0.001 0.01 0.1

error in L2 norm

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8

Accuracy on general meshes, INS

Compare the accuracy obtained on the Kovasznay flow problem with physical and reference frame dG discretizations.

INS dG discretization proposed by Di Pietro and Ern, [Math. Comp., 2010].

Randomly distorted mesh sequence Element subdivision mesh sequence

Accuracy on general meshes, INS, physical frame

for the velocity error in L2 norm for pressure error in L2 norm

Randomly distorted Element subdivision

10 100

sqrt(number of elements)

1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=1 p, k=1 u, k=2 p, k=2 u, k=3 p, k=3 u, k=4 p, k=4 u, k=5 p, k=5 u, k=6 p, k=6 u, k=7 p, k=7 u, k=8 p, k=8 u, k=9 p, k=9 u, k=10 p, k=10 10 100

sqrt(number of elements)

1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=1 p, k=1 u, k=2 p, k=2 u, k=3 p, k=3 u, k=4 p, k=4 u, k=5 p, k=5 u, k=6 p, k=6 u, k=7 p, k=7 u, k=8 p, k=8 u, k=9 p, k=9 u, k=10 p, k=10

Physical frame dG discretizations provide optimal approximation properties

• ver the randomly distorted and the element subdivision mesh sequence

Convergence rates

Accuracy on general meshes, INS, reference frame

10 100

sqrt(number of elements)

1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=1 u, k=2 u, k=3 u, k=4 u, k=5 u, k=6 u, k=7 u, k=8 u, k=9 u, k=10 10 100

sqrt(number of elements)

1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

p, k=1 p, k=2 p, k=3 p, k=4 p, k=5 p, k=6 p, k=7 p, k=8 p, k=9 p, k=10

Element subdivision mesh sequence

velocity pressure Convergence rates velocity pressure

10 100

sqrt(number of elements)

1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=1 u, k=2 u, k=3 u, k=4 u, k=5 u, k=6 u, k=7 u, k=8 u, k=9 u, k=10 u, k=11 10 100

sqrt(number of elements)

1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

p, k=1 p, k=2 p, k=3 p, k=4 p, k=5 p, k=6 p, k=7 p, k=8 p, k=9 p, k=10 p, k=11

Randomly distorted mesh sequence

4 4

velocity pressure Convergence rates velocity pressure

Accuracy on general meshes, INS, reference frame

10 100

sqrt(number of elements)

1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=1 u, k=2 u, k=3 u, k=4 u, k=5 u, k=6 u, k=7 u, k=8 u, k=9 u, k=10 10 100

sqrt(number of elements)

1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

p, k=1 p, k=2 p, k=3 p, k=4 p, k=5 p, k=6 p, k=7 p, k=8 p, k=9 p, k=10

Element subdivision mesh sequence

velocity pressure Convergence rates velocity pressure

10 100

sqrt(number of elements)

1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=1 u, k=2 u, k=3 u, k=4 u, k=5 u, k=6 u, k=7 u, k=8 u, k=9 u, k=10 u, k=11 10 100

sqrt(number of elements)

1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

p, k=1 p, k=2 p, k=3 p, k=4 p, k=5 p, k=6 p, k=7 p, k=8 p, k=9 p, k=10 p, k=11

Randomly distorted mesh sequence

4 4

velocity pressure Convergence rates velocity pressure

Good mesh generators

Accuracy comparison, element subdivision mesh sequence

Physical frame discretizations provide the best accuracy for a given number of dofs. The errors in L2 norm are up to fourth

• rder of magnitude smaller at the highest

polynomial degrees.

2 4 6 8 10

polynomial degree

1 1.5 2 2.5 3 3.5 4

CPU Time PFD/RFD

Matrix assembly Total time Linear system solution

Physical frame discretizations are more expensive due to numerical integration and basis function computation. The total simulation time is comparable since linear system solution times dominate assembly times.

10 100

sqrt(number of elements)

1x10-11 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

error in L2 norm

u, k=2 u, k=2 u, k=4 u, k=5 u, k=5 u, k=7 u, k=10 u, k=10 Reference Frame Physical Frame

1 10 100

Total Time (s)

1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3

L2 Velocity Error

RFD, k=6 PFD, k=6 RFD, k=8 PFD, k=8 RFD, k=10 PFD, k=10 0.001 0.01 0.1 1 10 100 1000

Total Time (s)

1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

L2 Velocity Error

RFD, k=1 PFD, k=1 RFD, k=2 PFD, k=2 RFD, k=4 PFD, k=4

Accuracy comparison, element subdivision mesh sequence

0.01 0.1 1 10

Matrix Assembly Time (s)

1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

L2 Pressure Error

RFD, k=1 PFD, k=1 RFD, k=2 PFD, k=2 RFD, k=4 PFD, k=4 RFD, k=6 PFD, k=6

Physical frame discretizations provide the best accuracy for a given computational cost.

Total CPU time

0.01 0.1 1 10

Matrix Assembly Time (s)

1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1

L2 Velocity Error

RFD, k=1 PFD, k=1 RFD, k=2 PFD, k=2 RFD, k=4 PFD, k=4 RFD, k=6 PFD, k=6

Physical frame discretizations are competitive in terms

• f accuracy for a given

computational cost.

Assembly time

Agglomerated elements

Physical frame basis functions can be defined on very general elements

• btained by agglomeration of sub-elements, see [Bassi et al, JCP

, 2011].

L-shaped element composed

• f rectangular sub-elements
• diagonal mass matrix
• shape independent cardinality
• hierarchic
• suitable for high-order

Fourth degree L2-orthonormal basis function modes

Basis function properties Example

Accuracy on agglomerated meshes, INS, Kovasznay flow

k=10 velocity solution, 16 elements mesh.

10

sqrt(number of elements)

1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 0.01 0.1 1

error in L2 norm

u, k=1 p, k=1 u, k=2 p, k=2 u, k=3 p, k=3 u, k=4 p, k=4 u, k=5 p, k=5 u, k=6 p, k=6 u, k=7 p, k=7 u, k=8 p, k=8 u, k=9 p, k=9 u, k=10 p, k=10 64

Agglomerated mesh sequence, 16, 64 and 256 elements (100 sub-cells per element).

Agglomerated elements, try to avoid high-order meshing.

500 elements mesh generated agglomerating (parMETIS) a 180k 10-node tetrahedrons mesh. k=7 Solution, Re400 500 agglomerated elements

Agglomerated elements, try to avoid high-order meshing. k=1, 180k tet mesh, 2.88M dofs k=5, 500 elem mesh, 112k dofs k=2, 180k tet mesh, 7.2M dofs k=7, 500 elem mesh 240k dofs.

Conclusion and future works

Physical frame discretizations ensure that the accuracy is maintained on general meshes. Agglomerated elements meshes simplify the work-flow required to perform high-order accurate discretizations.

• Standard low order elements and standard mesh generators can be employed.
• The mesh cardinality is independent from the boundary approximation.
• Better agglomeration strategies.
• Efficient quadrature rules on agglomerated elements.
• Efficient solution strategies based on h or p-multigrid.

To be effective in production runs we still need

All the results have been generated with open-source software, http://spafedte.github.com/ SpaFEDTe, a Template based C++ library for creating Discontinuous Finite Element Spaces