Introduction to geodynamic modelling - GitHub Pages · Introduction to geodynamic modelling Introduction to DOUAR David Whipp and Lars Kaislaniemi Department of Geosciences and Geography,

Intro to geodynamic modelling www.helsinki.fi/yliopisto May 20, 2018

Introduction to geodynamic modelling

Introduction to DOUAR

David Whipp and Lars Kaislaniemi Department of Geosciences and Geography, Univ. Helsinki

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www.helsinki.fi/yliopisto May 20, 2018Intro to geodynamic modelling

Goals of this lecture

• Introduce DOUAR, a 3D thermomechanical numerical modelling program for creeping flows

• Present some of the important features in DOUAR that will be relevant for our use

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DOUAR in a nutshell

• DOUAR (Braun et al., 2008) is a 3D finite-element code for modelling geodynamic processes

• It is designed to be run efficiently in parallel on computer clusters to be able to solve 3D problems at high spatial resolution

• We don’t have time for a complete description of the numerical and computational aspects of DOUAR, but we’ll see the highlights

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DOUAR is the word for Earth in the Breton language


Physics of DOUAR

• DOUAR calculates the flow of a highly viscous flow using the Stokes equation, which we have previously seenwhere 𝜂 is the fluid viscosity, 𝒗 is the fluid velocity, 𝑝 is pressure, 𝜌 is density, and 𝑔 is the acceleration due to gravity.

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Incompressible flow, almost

• It is further assumed that the fluid is nearly incompressible

• In an incompressible fluid the divergence of the velocity field is zero

• In DOUAR, the slight compressibility of the fluid is used to determine fluid pressure (eliminating it from the Stokes equation) using the penalty method where 𝜆 is the penalty factor (typically 8 orders of magnitude larger than the shear viscosity)

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Rheologies

• Materials in DOUAR are either viscous or plastic (no elasticity)

• Nonlinear viscosity is modelled using the equation for temperature-dependent nonlinear viscosity where 𝜂0 is the viscosity pre-factor, 𝜀 is the strain rate, 𝑛 is the power law exponent, 𝑄 is the activation energy, 𝑅 is the universal gas constant, and 𝑇 is temperature in Kelvins

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⌘ = ⌘0"̇1/(n�1)

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.


Rheologies

• There are several options for different plasticity criteria

• The Mohr-Coulomb criterion is what we use most often where 𝜏 is the shear stress, 𝑐 is the cohesion, 𝜎n is the normal stress, and 𝜙 is the internal angle of friction

7

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Thermal model

• The full 3D advection-diffusion equation with heat production is solved in DOUAR where 𝑐 is the heat capacity, 𝑇 is temperature, 𝑡 is time, 𝑘 is the

thermal conductivity, and 𝐻 is the heat production per unit mass.

8

⇢c

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Numerical approach

• The finite-element mesh used in DOUAR is Eulerian and based on the octree division of space

• An Eulerian mesh is one that is fixed in space with respect to the fluid flowing within/through it

• The octree division of space is based on subdivisions of a unit cube

9

J. Braun et al. / Physics of the Earth and Planetary Interiors 171 (2008) 76–91 79

Fig. 1. Example of a simple octree discretization of the unit cube. The unit cube is divided in eight sub-cubes, which can be arbitrarily divided into eight sub-sub-cubes,and so on. The sub-cubes that remain undivided at the end of the construction of the octree are called leaves which are used here as finite elements with which the partialdifferential equations are solved.

of the small elements do not exist in the adjacent large elements.These are called ‘bad faces’ that are dealt with by imposing linearconstraints (Webb, 1990) as shown in Fig. 2.

The q1–p0 elements are known to be affected by the presenceof a “checkerboard” mode in the pressure field (Bathe, 1982). Theintroduction of the linear constraints into the set of finite elementequations may also contribute to these unwanted oscillations. Tominimize them, we “smooth” the pressure field by performing adouble interpolation of the elemental pressure onto the nodes,and then back onto the elements. The element-to-node interpola-tion is performed by averaging the elemental values from elementscommon to each node; the node-to-element interpolation is per-formed by averaging the nodal values element-by-element. Thismethod is not only very efficient but produces a smoothing ofthe pressure that is adapted to the local density of the octree andcan be shown to be equivalent to a least-square smoothing of thepressure.

Fig. 2. The interface between two sets of leaves of different level is called a ‘bad face’.These bad faces contain nodes that belong to the smaller elements on one side ofthe face and not to the larger element on the other side of the face. These nodes arefilled in black and grey on the figure. Using the finite element method, one can onlysolve for velocity components and temperature on nodes that belong to elementson both sides of the face (the white nodes). To obtain values at the incompatiblenodes, one needs to impose additional linear constraints that constrain the solutionat the grey mid-side nodes to be the mean of the two adjacent red nodes, and thesolution at the central black node to be the mean of the four corners nodes.

Octrees are very simple and memory-efficient entities that canbe built as a single integer array containing, for each cube of theoctree, the address in the array of the first of its eight ‘childrencubes’. In the scheme we have developed here, when a cube isnot divided, it becomes a leaf to which a name/number is associ-ated and is stored in the octree integer array as a negative number(to indicate that it corresponds to a leaf number and not a child’saddress). This scheme is memory efficient (most octrees are only afew kilobytes in size) but requires additional operations when per-forming operations on the octree. However, most of the operationscommonly needed in the construction of a finite element problemare done with great efficient when using the octree storage schemedescribed above. Most global operations (i.e. those affecting all theleaves of an octree) require only ∼N arithmetic or conditional oper-ations. For example, for an octree of maximum depth Lmax (level ofthe smallest leaf or finite element in the octree) and made of Nleaves:

• to create a leaf at level L around a point of known coordinates;this requires L conditional statements;

• to locate a point of known coordinates, i.e. to find thename/number of the leaf it belongs to; we will call this a ‘location’operation and is achieved through Lmax conditional statementsfor each of the point coordinates;

• to determine the size of all leaves/elements; this operationinvolves ∼N conditional statements

• to find the list of neighbouring leaves/elements; this operationinvolves ∼26N location operations;

• to interpolate a field known at the nodes of an octree; this oper-ation involves a location and a trilinear interpolation operationper interpolation;

• to unite two octrees; this involves checking the depth of oneoctree for each leaf of the other octree and, if necessary, creatinga leaf;

• to smooth an octree (see Fig. 3); this requires ∼6N location andconditional operations.

When using the simple scheme described above to store theoctree, one operation becomes however relatively costly: to findthe connectivity matrix between nodes and leaves/elements. Thisis done here by first numbering the nodes in a redundant manner,i.e. by giving to each element a set of eight nodes, regardless of con-nectivities between the elements and the possibility that a singlenode is given many different names/numbers in the sequence, i.e.

Braun et al., 2008


Numerical approach

• The unit cube is at the octree level 0 resolution, meaning that there are 20 elements along the width of the cube

• A mesh at octree level 6 would have 26, or 64 x 64 x 64 elements

• The initial octree resolution is an important setting in the input file

10

J. Braun et al. / Physics of the Earth and Planetary Interiors 171 (2008) 76–91 79

Fig. 1. Example of a simple octree discretization of the unit cube. The unit cube is divided in eight sub-cubes, which can be arbitrarily divided into eight sub-sub-cubes,and so on. The sub-cubes that remain undivided at the end of the construction of the octree are called leaves which are used here as finite elements with which the partialdifferential equations are solved.

of the small elements do not exist in the adjacent large elements.These are called ‘bad faces’ that are dealt with by imposing linearconstraints (Webb, 1990) as shown in Fig. 2.

The q1–p0 elements are known to be affected by the presenceof a “checkerboard” mode in the pressure field (Bathe, 1982). Theintroduction of the linear constraints into the set of finite elementequations may also contribute to these unwanted oscillations. Tominimize them, we “smooth” the pressure field by performing adouble interpolation of the elemental pressure onto the nodes,and then back onto the elements. The element-to-node interpola-tion is performed by averaging the elemental values from elementscommon to each node; the node-to-element interpolation is per-formed by averaging the nodal values element-by-element. Thismethod is not only very efficient but produces a smoothing ofthe pressure that is adapted to the local density of the octree andcan be shown to be equivalent to a least-square smoothing of thepressure.

Fig. 2. The interface between two sets of leaves of different level is called a ‘bad face’.These bad faces contain nodes that belong to the smaller elements on one side ofthe face and not to the larger element on the other side of the face. These nodes arefilled in black and grey on the figure. Using the finite element method, one can onlysolve for velocity components and temperature on nodes that belong to elementson both sides of the face (the white nodes). To obtain values at the incompatiblenodes, one needs to impose additional linear constraints that constrain the solutionat the grey mid-side nodes to be the mean of the two adjacent red nodes, and thesolution at the central black node to be the mean of the four corners nodes.

Octrees are very simple and memory-efficient entities that canbe built as a single integer array containing, for each cube of theoctree, the address in the array of the first of its eight ‘childrencubes’. In the scheme we have developed here, when a cube isnot divided, it becomes a leaf to which a name/number is associ-ated and is stored in the octree integer array as a negative number(to indicate that it corresponds to a leaf number and not a child’saddress). This scheme is memory efficient (most octrees are only afew kilobytes in size) but requires additional operations when per-forming operations on the octree. However, most of the operationscommonly needed in the construction of a finite element problemare done with great efficient when using the octree storage schemedescribed above. Most global operations (i.e. those affecting all theleaves of an octree) require only ∼N arithmetic or conditional oper-ations. For example, for an octree of maximum depth Lmax (level ofthe smallest leaf or finite element in the octree) and made of Nleaves:

• to create a leaf at level L around a point of known coordinates;this requires L conditional statements;

• to locate a point of known coordinates, i.e. to find thename/number of the leaf it belongs to; we will call this a ‘location’operation and is achieved through Lmax conditional statementsfor each of the point coordinates;

• to determine the size of all leaves/elements; this operationinvolves ∼N conditional statements

• to find the list of neighbouring leaves/elements; this operationinvolves ∼26N location operations;

• to interpolate a field known at the nodes of an octree; this oper-ation involves a location and a trilinear interpolation operationper interpolation;

• to unite two octrees; this involves checking the depth of oneoctree for each leaf of the other octree and, if necessary, creatinga leaf;

• to smooth an octree (see Fig. 3); this requires ∼6N location andconditional operations.

When using the simple scheme described above to store theoctree, one operation becomes however relatively costly: to findthe connectivity matrix between nodes and leaves/elements. Thisis done here by first numbering the nodes in a redundant manner,i.e. by giving to each element a set of eight nodes, regardless of con-nectivities between the elements and the possibility that a singlenode is given many different names/numbers in the sequence, i.e.

Braun et al., 2008


The octree division of space

• As an example, here is a face of a DOUAR model (2D, not full cube) at octree level 2

• This is 2 subdivisions of the unit cube (red)

11

Leaf size and levels (3)

level = 2 leaf size=2−2=0.25

CT. Feb. 2007 – p. 41/190

Level 0

Level 1

Level 2


Other aspects of DOUAR

• (continued on whiteboard)

12


References

Braun, J., Thieulot, C., Fullsack, P., DeKool, M., Beaumont, C. and Huismans, R., 2008. DOUAR: A new three-dimensional creeping flow numerical model for the solution of geological problems. Physics of the Earth and Planetary Interiors, 171(1-4), pp.76-91.

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Documents

Introduction to geodynamic modelling - GitHub Pages · Introduction to geodynamic modelling Introduction to DOUAR David Whipp and Lars Kaislaniemi Department of Geosciences and Geography,