In
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
, numerical modeling is a widely applied technique to tackle complex geological problems by computational simulation of geological scenarios.
Numerical modeling uses
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
s to describe the physical conditions of geological scenarios using numbers and equations.
Nevertheless, some of their equations are difficult to solve directly, such as
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. With numerical models, geologists can use methods, such as
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
s, to approximate the solutions of these equations. Numerical experiments can then be performed in these models, yielding the results that can be interpreted in the context of geological process.
Both qualitative and quantitative understanding of a variety of geological processes can be developed via these experiments.
Numerical modelling has been used to assist in the study of
rock mechanics Rock mechanics is a theoretical and applied science of the mechanical behavior of rock and rock masses; compared to geology, it is that branch of mechanics concerned with the response of rock and rock masses to the force fields of their physical env ...
, thermal history of rocks, movements of tectonic plates and the Earth's mantle. Flow of fluids is simulated using numerical methods, and this shows how
groundwater
Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidate ...
moves, or how motions of the molten outer core yields the geomagnetic field.
History
Prior to the development of numerical modeling,
analog modeling, which simulates nature with reduced scales in mass, length, and time, was one of the major ways to tackle geological problems,
for instance, to model the formation of
thrust belts. Simple analytic or semi-analytic mathematical models were also used to deal with relatively simple geological problems quantitatively.
In the late 1960s to 1970s, following the development of
finite-element methods in solving
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
problems for
civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
, numerical methods were adapted for modeling complex geological phenomena,
for example,
folding
Fold, folding or foldable may refer to:
Arts, entertainment, and media
* ''Fold'' (album), the debut release by Australian rock band Epicure
* Fold (poker), in the game of poker, to discard one's hand and forfeit interest in the current pot
*Abov ...
and
mantle convection
Mantle convection is the very slow creeping motion of Earth's solid silicate mantle as convection currents carrying heat from the interior to the planet's surface.
The Earth's surface lithosphere rides atop the asthenosphere and the two form ...
. With advances in computer technology, the accuracy of numerical models has been improved.
Numerical modeling has become an important tool for tackling geological problems,
especially for the parts of the Earth that are difficult to observe directly, such as the
mantle and
core
Core or cores may refer to:
Science and technology
* Core (anatomy), everything except the appendages
* Core (manufacturing), used in casting and molding
* Core (optical fiber), the signal-carrying portion of an optical fiber
* Core, the central ...
. Yet analog modeling is still useful in modeling geological scenarios that are difficult to capture in numerical models, and the combination of analog and numerical modeling can be useful to improve understanding of the Earth's processes.
Components
A general numerical model study usually consists of the following components:
# Mathematical model is a simplified description of the geological problem, such as equations and boundary conditions.
These governing equations of the model are often
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s that are difficult to solve directly since it involves the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, for example, the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
.
# Discretization methods and numerical methods convert those governing equations in the mathematical models to discrete equations.
These discrete equations can approximate the solution of the governing equations.
Common methods include the
finite element
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
,
finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
, or
finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
that subdivide the object of interest into smaller pieces (element) by mesh. These discrete equations can then be solved in each element numerically.
The
discrete element method
A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, t ...
uses another approach, this method reassembling the object of interest from numerous tiny particles. Simple governing equations are then applied to the interactions between particles.
# Algorithms are computer programs that compute the solution using the idea of the above numerical methods.
# Interpretations are made from the solutions given by the numerical models.
Properties
A good numerical model usually has some of the following properties:
* Consistent: Numerical models often divide the object into smaller elements. If the model is consistent, the result of the numerical model is nearly the same as what the mathematical model predicts when the element size is nearly zero. In other words, the error between the discrete equations used in the numerical model and the governing equations in the mathematical model tends to zero when the space of the mesh (size of element) becomes close to zero.
*
Stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
: In a stable numerical model, the error during the computation of the numerical methods does not amplify.
The error of an unstable model will stack up quickly and lead to an incorrect result. A ''stable'' and ''consistent'' numerical model has the same output as the exact solution in the mathematical model when the spacing of the mesh (size of element) is extremely small.
* Converging: The output of the numerical model is closer to the actual solution of the governing equations in the mathematical models when the spacing of mesh (size of element) reduces, which is usually checked by carrying out numerical experiments.
* Conserved: The physical quantities in the models, such as mass and momentum, are conserved.
Since the equations in the mathematical models are usually derived from various conservation laws, the model result should not violate these premises.
* Bounded: The solution given by the numerical model has reasonable physical bounds with respect to the mathematical models, for instance mass and volume should be positive.
* Accurate: The solution given by the numerical models is close to the real solution predicted by the mathematical model.
Computation
The following are some key aspects of ideas in developing numerical models in geology. First, the way to describe the object and motion should be decided (
kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
description). Then, governing equations that describe the geological problems are written, for example, the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
s describe the flow of heat in a system. Since some of these equations cannot be solved directly, numerical methods are used to approximate the solution of the governing equations.
Kinematic descriptions
In numerical models and mathematical models, there are two different approaches to describe the motion of matter: Eulerian and Lagrangian.
In geology, both approaches are commonly used to model fluid flow like mantle convection, where an Eulerian grid is used for computation and Lagrangian markers are used to visualize the motion.
Recently, there have been models that try to describe different parts using different approaches to combine the advantages of these two approaches. This combined approach is called the arbitrary Lagrangian-Eulerian approach.
Eulerian
The Eulerian approach considers the changes of the physical quantities, such as mass and velocity, of a ''fixed location'' with time.
It is similar to looking at how river water flows past a bridge. Mathematically, the physical quantities can be expressed as a function of location and time. This approach is useful for fluid and homogeneous (uniform) materials that have no natural boundary.
Lagrangian
The Lagrangian approach, on the other hand, considers the change of physical quantities, such as the volume, of ''fixed elements'' of matter over time.
It is similar to looking at a certain collection of water molecules as they flow downstream in a river. Using the Lagrangian approach, it is easier to follow solid objects which have natural boundary to separate them from the surrounding.
Governing equations
Following are some basic equations that are commonly used to describe physical phenomena, for example, how the matter in a geologic system moves or flows and how heat energy is distributed in a system. These equations are usually the core of the mathematical model.
Continuity equation
The
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
is a mathematical version of stating that the geologic object or medium is continuous, which means no empty space can be found in the object.
This equation is commonly used in numerical modeling in geology.
One example is the continuity equation of mass of fluid. Based on the law of ''conservation of mass'', for a fluid with density
at position
in a fixed volume
of fluid, the rate of change of mass is equal to the outward fluid flow across the boundary
:
where
is the volume element and
is the velocity at
.
In Lagrangian form:
In Eulerian form:
This equation is useful when the model involves continuous fluid flow, like the mantle is over geological time scales.
Momentum equation
The momentum equation describes how matter moves in response to force applied. It is an expression of
Newton's second law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
.
Consider a fixed volume
of matter. By the law of ''conservation of momentum'', the rate of change of volume is equal to:
* external force
applied on the element
* plus normal stress and shear stress applied on the surface
bounding the element
* minus the momentum moving out of the element on that surface
where
is the volume element,
is the velocity.
After simplifications and integrations, for any volume
, the Eulerian form of this equation is:
Heat equation
The heat equations describe how
heat energy
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
flows in a system.
From the law of conservation of energy, the rate of change of energy
of a fixed volume
of mass is equal to:
* work done at the boundary
* plus work done by external force
in the volume
* minus heat
conduction
Conductor or conduction may refer to:
Music
* Conductor (music), a person who leads a musical ensemble, such as an orchestra.
* Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas
* Conduction, a type of structured f ...
across boundary
* minus heat
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
across boundary
* plus heat produced internally
Mathematically:
where
is the volume element,
is the velocity,
is the temperature,
is the
conduction coefficient and
is the rate of heat production.
Numerical methods
Numerical methods are techniques to approximate the governing equations in the mathematical models.
Common numerical methods include
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
,
spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis functio ...
,
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
, and
finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
. These methods are used to approximate the solution of governing
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s in the mathematical model by dissecting the domain into meshes or grids and applying simpler equations to individual elements or nodes in the mesh.
The
discrete element method
A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, t ...
uses another approach. The object is considered an assemblage of small particles.
Finite element method
The
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
subdivides the object (or domain) into smaller, non-overlapping elements (or subdomains) and these elements are connected at the nodes. The solution for the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s are then approximated by simpler element equations, usually
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s.
Then these element equations are combined into equations for the entire object, i.e. the contribution of each element is summed up to model the response of the whole object.
This method is commonly used to solve mechanical problems.
The following are the general steps of using the finite element method:
# Select the element type and subdivide the object. Common
element types include triangular, quadrilateral, tetrahedral, etc.
Different types of elements should be chosen for different problems.
# Decide the function of displacement. The function of displacement governs how the elements move. Linear, quadratic, or
cubic polynomial
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
functions are commonly used.
# Decide the displacement-strain relation. The displacement of the element changes or deforms the element's shape in what is technically called
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
. This relation calculates how much strain the element experienced due to the displacement.
# Decide the strain-stress relation. The deformation of the element induces
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
to the element, which is the
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
applied to the element. This relation calculates the amount of stress experienced by the element due to the strain. One of the examples of this relation is
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
.
# Derive equations of stiffness and stiffness matrix for elements. The stress also causes the element to deform; the
stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
(the rigidity) of the elements indicates how much it will deform in response to the stress. The stiffness of the elements in different directions is represented in
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
form for simpler operation during calculation.
# Combine the element equations into global equations. The contributions of every element are summed up to a set of equations that describe the whole system.
# Apply boundary conditions. The predefined conditions at the boundary, such as temperature, stress, and other physical quantities are introduced to the boundary of the system.
# Solve for displacement. As time evolves, the displacement of the elements are solved step by step.
# Solve for strains and stress. After the displacement is calculated, the strains and stress are computed using the relations in steps 3 and 4.
Spectral method
The
spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis functio ...
is similar to the finite element method.
The major difference is that spectral method uses
basis functions
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be repres ...
, possibly by using a
Fast Fourier Transformation (FFT) that approximates the function by the sum of numerous simple functions.
These kinds of basis functions can then be applied to the whole domain and approximate the governing
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s.
Therefore, each calculation takes the information from the whole domain into account while the finite element method only takes the information from the neighborhood.
As a result, the spectral method converges exponentially and is suitable for solving problems involving a high variability in time or space.
Finite volume method
The
finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
is also similar to the finite element method. It also subdivides the object of interest into smaller volumes (or elements), then the physical quantities are solved over the control volume as fluxes of these quantities across the different faces.
The equations used are usually based on the conservation or balance of physical quantities, like mass and energy.
The finite volume method can be applied on irregular meshes like the finite element method. The element equations are still physically meaningful. However, it is difficult to get better accuracy, as the higher order version of element equations are not well-defined.
Finite difference method
The
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
approximates
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s by approximating the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
with a
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, which is the major method to solve
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s.
Consider a function
with single-valued derivatives that are continuous and finite functions of
, according to
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
:
and
Summing up the above expressions:
Ignore the terms with higher than 4th power of
, then: