Numerical Modeling & FLAC3D Introduction.pptx

Numerical Modeling &
FLAC3D
Visvesvaraya National Institute of Technology Nagpur
By
Kothakonda Maharshi
DT21MIN001
Ph.D in Mining Engineering

Outline
• Numerical Modeling and its types
• Finite Volume Method and its comparison
• How to decide a Modeling Technique
• 2D vs 3D Modeling
• FLAC3D Introduction and its features
• Mesh, Types of Meshes and Mesh Quality
• Mesh Generation Tools
FLAC3D (Explicit Continuum Modeling of Non-linear Material Behavior in 3D) Presentation by Maharshi (DT21MIN001) 2

Why Numerical Modeling is used
?

Why Numerical Modeling ?
• The best way to solve a physical problem governed by differential equations is to
obtain a closed form analytical solution.
• Unfortunately there are many practical situations where the analytical solution is
difficult to obtain or analytical solution does not exist.
• This is because in order to obtain analytical solution, the shape of the physical
domain must be known in the mathematical form. If the shape of the domain is
irregular, so that no mathematical representation can be made. Then it is
impossible to solve the problem using analytical method.
• Even if the shape of the domain is regular, many a times the governing equations
of the problem may be nonlinear and makes the problem more complex to solve.
• There are several procedures to obtain an approximate numerical solution to a
partial differential equation.

Different Numerical Methods
• In broad sense, numerical methods can be classified into continuum and discontinuum.
The concept of continuum and discontinuum are not absolute but relative and problem
specific depending specially on geometric configuration.
• A continuum model is a representation of a structure or a process of gradual and
uninterrupted change in its elements between two distinctive points defined by a
particular measure.
• There are several procedures to obtain an approximate numerical solution to a partial
differential equation.
• Some of the popular methods are:-
Finite Element Method (FEM)
Finite Difference Method (FDM)
Finite Volume Method (FVM)

Flow chart depicting the procedure followed
to solve a problem by FVM, FDM & FEM
Fig: The most commonly used numerical methods and their essential differences

Finite Volume Discretization Approaches
• The FVM divides a geometrically arbitrary domain
into a finite number of elements (a structured or
unstructured mesh), subsequently used to build finite or
control volumes (a dual mesh).
• The discretization of the domain into control volumes
can be performed by adopting a vertex-centred
approach (where each node of the mesh is the centre
of a finite volume, whose boundaries are obtained by
connecting the centroids of each element and
the midpoints of each element edge), or by a cell-
centred approach (where control volumes coincide
with elements), as depicted in Figure for the two-
dimensional case.
Fig: Mesh and dual mesh in vertex-centred FVM (a, b) and cell-centred
FVM (c, d). Control volumes are defined by the grey-coloured areas [4]

FDM vs FVM
Finite Difference Method (FDM)
• Differential form of governing
equations
• Domain is discretized into finite
discrete points
• Defined on a regular/structured grid
• Regular grids are suitable for very
large scale simulations
• Solution will be obtained on nodal
points
Finite Volume Method (FVM)
FLAC3D (Explicit Continuum Modeling of Non-linear Material
Behavior in 3D) Presentation by Maharshi (DT21MIN001)
8
• Integral form of governing equations
• Domain is discretized into finite
volumes or cells
• Defined on any irregular/unstructured
mesh
• Irregular grids are suitable for
complex geometries
• Solution will be obtained on centre of
each cell

How to decide Modeling Technique ?
Method Scheme Time for
Convergence
Applicability Error Geometry
FDM Difference
Equations and
uses Taylor’s
Expansion
Fast Very good for
Higher order
PDE’s
more Regular and
Large-scale
Problems
FVM Integral form
and uses
Divergence
Theorem
Slow Good (up to
second order
PDE’s)
less Regular/
Irregular and
Complex
geometries
FEM stiffness
matrices and
Uses basic
functions
Slow Good (up to
second order
PDE’s)
less Regular/
Irregular and
Complex
geometries

Lagrangian vs Eulerian Approach
• The choice of either a Lagrangian or an Eulerian viewpoint depends on the
physical problem under investigation.
• Normally, the Eulerian description is used in fluid mechanics because particles
have very large, complicated motions relative to each other and, in most cases, the
interest is concentrated in the condition of flow in a fixed region of space.
• While in solid mechanics, the Lagrangian system is normally employed because it
provides an easier method of identifying paths of particles which is important in
problems involving deformations of boundaries which are often not known before.

2D vs 3D Modeling
• All real-world stress and strain states are three-dimensional. In many cases,
however, we can make assumptions that greatly simplify the analysis, without
greatly affecting the result.
• 2D approaches can used only if the problem fulfils the plain strain criteria i.e. if
the out-of-plain strain resulting from the model perturbation is zero or uniform.
• The simplest example of this is a straight tunnel in a homogenous rock mass
• If the plain strain criteria is not fulfilled (i.e. for a tunnel intersection or curve) a
3D model is required.

What is FLAC3D ?
• FLAC3D is a three-dimensional explicit Lagrangian finite-volume program.
• It is a continuum numerical modeling software for geotechnical analyses of
soil, rock, ground water and ground supports.
• It is used mainly in civil, mining, geotechnical and petroleum engineering
applications.

Audio Clip from ITASCA
Youtube Link: https://www.youtube.com/watch?v=dCkmGbQhwVI&t=40s

Features of FLAC3D
• FLAC3D is mostly a command driven program (i.e., FISH and Python scripting
languages) which makes it more powerful.
 Parametric Studies
 Automating model sequences
 Modification of Physics
• FLAC3D features accelerated 3D graphics that allow for rapid model
visualization.
• Extensive plotting capabilities including easy mechanisms for constructing
animations (videos) are available to facilitate pre and post processing.
• Dynamic Modeling and Thermal Analysis.
• 26 Mechanical and 9 Creep Constitutive Models built-in.
• Can create your own constitutive model.

Features of FLAC3D
• Model Building: Extrusion Pane, Building Blocks and Imported geometric surfaces.
• Densification of zones close to imported surfaces or interfaces or at a desired
location.
• Import the geometric meshes of a physical domain from third-party CAD softwares
like Rhino, AutoCAD, etc.
• Can create structural elements like Beams, Cables, Piles, Shells, Liners & Geogrids.
• Can create interfaces like Faults, Joints.
• Can create water table.
• Time varying loads can be applied.
• Can view results as calculation progresses.

Numerical Formulation of FLAC3D
The method of solution in FLAC3D is characterized by the following three approaches:
• Finite volume approach (First-order space and time derivatives of a variable are
approximated by finite volumes assuming linear variations of the variable over finite
space and time intervals, respectively);
• Discrete-model approach (The continuous medium is replaced by a discrete equivalent
one in which all forces involved (applied and interactive) are concentrated at the nodes of
a three-dimensional mesh used in the medium representation); and
• Dynamic-solution approach (The inertial terms in the equations of motion are used as
numerical means to reach the equilibrium state of the system under consideration).

Sign Conventions used in FLAC3D
Itasca software follows mechanical engineering practices:
• Direct Stress: Positive stresses indicate tension; negative stresses indicate compression
• Shear Stress: Positive shear stress points in the positive direction of the coordinate axis
• Direct Strain: Positive strain indicates extension; negative strain indicates compression
• Shear Strain: Follows the convention of shear stress

What is a Mesh ?
• The finite volume grid is an assemblage of one or more finite volume zones across
the physical region that is being analysed. Another term for grid is mesh.
• Gridpoints are associated with the corners of the finite volume zones. There are
four, five, six, seven, or eight gridpoints associated with each polyhedral zone,
depending on the zone shape.

Mesh Types
Meshes are often categorized as: structured, unstructured,
or hybrid meshes.
• Structured Mesh: Structured meshes are identified by
regular connectivity.
• For example, a quadrilateral mesh in 2D is structured if
each internal node is joined to 4 neighbouring
quadrilaterals, forming a regular array of elements.
• In 3D, a structured hexahedral grid has each internal
node connected to 8 elements. Structured meshes
typically have well-shaped elements.
Fig: Structured Mesh of a domain [2].

Mesh Types
• Unstructured Mesh: An unstructured grid is
identified by irregular connectivity.
• Hybrid mesh: usually refers to a mesh that
contains a combination of structured and
unstructured meshes.
• In general, structured meshes offer
simplicity and easy data access, while
unstructured meshes offer more convenient
mesh adaptivity and a better fit to
complicated domains.
Fig: Unstructured Mesh of a domain [2].

Mesh Types
• Meshes are also categorized as: conformal
and non-conformal meshes.
• Conformal: A conformal mesh is a mesh
where each zone in the mesh, completely
shares a face with the neighbour (it should
also share the edges and nodes of that face as
well).
• In conformal mesh, the nodes at the contact
region are exactly one-one match, such as the
red nodes and green nodes in the conform
mesh plot.
Fig: Conformal Mesh of a 2D domain [5].

Mesh Types
• Non-Conformal: A non-conformal mesh includes
a hexahedral element sharing one of its
quadrilateral faces with more than one face of
adjoining element/zone.
• Another example includes the case where two
meshes might be laterally offset from each other,
where some of the nodes and edges do not match
at element boundaries (sometimes referred to as
"hanging nodes").
• FLAC3D is capable of handling both of these grid
types.
Fig: Non-conformal Mesh of a 2D domain [5].

What is a Octree Mesh ?
• Consider an initial structured
hexahedral grid. If we refine this grid
by subdividing hexahedrons (a hex is
subdivided into eight smaller
hexahedrons) in areas of interest.
• We continue subdividing the smaller
hexahedrons in our areas of interest up
to a certain level of refinement.
• If we then enforce the rule that an
element’s neighbour must be either ½,
1, or 2 times its size (by subdividing
further), we obtain a balanced octree
mesh.

What is a Octree Mesh ?
• With this method, the user initially creates a regular structured hexahedral grid
independent of the geometry and densifies this grid close to the features that need
to be modeled and analyzed.
• The level of densification will depend on the precision that is required.
• The densification process consists of dividing a hexahedron into eight
hexahedrons and repeating this until the desired zone size is achieved.
• The main advantage of the octree method is its ability to generate a mesh for
complex geometries very quickly.
• The main disadvantage is that the resulting mesh does not conform to the
geometry, but rather results in “stair step” boundaries that approximate the
modeled features, which is not acceptable for all types of problems.

Mesh Quality
• A good mesh is a mesh that allows you to solve your problem at the expected level
of accuracy within the time available for the project.
• Mesh quality is crucial for the stability, accuracy, and fast convergence of a
numerical simulation.
• There are numerous measures of mesh or element quality, such as element aspect
ratio, Jacobian, planarity of element faces, maximum and minimum angles at
corners, etc.

Mesh Metrics
• “Mesh Metrics” is one of the most useful features in
determining the correct shape and size of the elements.
• You can find a range of criteria for quality check of
your mesh like Aspect Ratio, Jacobian Ratio and
Skewness, etc.
Aspect Ratio
• The Aspect Ratio quantifies the quality of the
elements, where 1 is a perfectly shaped tetrahedral
element and the element shape is worse with a
higher Aspect Ratio.
• The aspect ratio is defined as the ratio of the shortest
length of the element to the longest length of the
element.
Fig: Showing different Aspect Ratio Elements

Mesh Metrics
Jacobian
• Jacobian Ratio is a measure of the deviation of a
given element from an ideally shaped element.
• The Jacobian value ranges from -1.0 to 1.0,
where 1.0 represents a perfectly shaped element.
Fig: Showing different Jacobian Ratio Elements

Mesh Metrics
Skewness
• Skewness is the Angular Measure of
Element quality with respect to the Angles
of Ideal Element Types.
• It is one of the Primary Qualities Measures
of FE Mesh. Skewness determines how
close to ideal (i.e., equilateral or equi-
angular) a face or cell is.
• The acceptable range of skewness is “0 to
0.5”.
Fig: Showing different Triangular and quad Elements

Important Aspect in Numerical Modeling
Convergence Analysis
• The most fundamental and accurate method for evaluating mesh quality is to
refine the mesh until a critical result, such as the maximum stress in a specific
location converges (i.e. it doesn’t change significantly with each refinement).
• Basically, the convergence tool increases the mesh density and checks the results
between each step. You can easily see how your results change depending on the
element quantity.
• The problem with this method is that it requires multiple remeshing and re-solving
operations. While this method is fine for simple models, it can be very time-
consuming for complex models. However, in Ansys you can easily perform this
operation automatically by using Convergence tool option.

Important Aspect in Numerical Modeling
Convergence Analysis
• An example is shown in Figure 1, where a 2D
bracket model is constrained at its top end and
subjected to a shear load at the edge on the lower
right. This generates a peak stress in the fillet, as
shown.
• The curve shows that as the mesh density increases,
the peak stress in the fillet increases.
• Ultimately, increasing the mesh density further
produces only minor increases in peak stress.
• In this case, an increase from 1134 elements per
unit area to 4483 elements per unit area yields only
a 1.5% increase in stress.

Mesh Generation Tools
• Mesh or Grid generation with FLAC3D involves adjusting and shaping the mesh
to fit the shape of the physical domain.
• Itasca offers a number of mesh generation tools. They may be divided into two
categories.
1. Using built-in meshing capabilities that are standard features with FLAC3D or
3DEC.
2. Using Rhino, CAD-based capabilities offered by Griddle or BlockRanger. Itasca
introduced a new powerful and easy-to-use automatic mesher called Griddle.
Griddle is accessible from the Rhino CAD-system and can output both all-
tetrahedral and hex-dominant meshes.

Introduction to FLAC3D Software?

References
[1] http://docs.itascacg.com/flac3d700/flac3d/docproject/source/flac3dhome.html
[2] Aissa, M., 2017. GPU-accelerated CFD simulations for turbomachinery design optimization (Doctoral
dissertation, Doctoral thesis, Delft University of Technology).
[3] Maneeratana, K., and A. Ivankovic. "Finite volume method for large deformation with linear hypoelastic
materials." In second international symposium on finite volumes for complex applications (FVCA II) for
complex applications II: problems and perspectives. Hermes Science Publication, University Duisburg,
Germany, pp. 459-466. 1999.
[4] Aleksendric, Dragan, and Pierpaolo Carlone. Soft Computing in the Design and Manufacturing of Composite
Materials: Applications to Brake Friction and Thermoset Matrix Composites. Woodhead Publishing, 2015.
[5] Chen, Lu, Xiaowei Zhou, Zhigao Huang, and Huamin Zhou. "Three-Dimensional Transient Finite Element
Cooling Simulation For Injection Molding Tools." (2021).
[6] https://www.youtube.com/watch?v=dCkmGbQhwVI&t=40s

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