FEATool Multiphysics
v1.10 Finite Element Analysis Toolbox |

Physics Modes

In **Equation** mode partial differential equations (PDEs) together with equation coefficients must be chosen to accurately describe the physical phenomena to be simulated. Furthermore, in **Boundary** mode suitable boundary conditions must be prescribed in order to account for how the model interacts with its surroundings (outside of the modeled geometry and grid).

After creating a suitable geometry and grid one can switch between equation and boundary modes by using the and mode buttons. The corresponding **Equation** and **Boundary Settings** dialog boxes will also automatically open. Each boundary and equation setting tab corresponds to a different physics mode present in the model and allow for specifying the equation and boundary coefficients, initial conditions, and finite element shape functions. The coefficients and boundary coefficients will vary depending on the chosen physics mode and are explained in the following sections. The equation settings also allow for different equation and initial conditions to be set on a per subdomain basis by selecting the subdomains in the left hand side **Subdomain:** listbox.

Moreover, a convenient way to set up models is to use **Model Constants and Expressions**. The button opens the corresponding dialog box where one can enter constants and define expressions. One entered they can be used in stead of entering numerical values in equation coefficients and postprocessing expressions. More space for coefficients and expressions can also be added with the **Add Row** button.

The FEATool GUI also makes it easy to add and couple multiphysics equations and complex expressions to your models.

After starting to work with a physics mode it is simple to add one or more other modes by going to the last tab, with a **+** plus sign, in the **Equation Settings** dialog box. There you can simply choose an additional physics mode from the drop down combo box, select the dependent variable names you want to use (or keep the default ones), and press **Add Physics >>>** to add the mode. The new physics mode will now show up as a new tab with its short abbreviated name on the tab handle. Once the new mode has been added you can switch between the modes by clicking on the corresponding tabs in the Equation and Boundary Settings dialog boxes.

The physics modes can be coupled by simply using the dependent variable names and derivatives in the coefficient expression dialog boxes. For example, the Navier-Stokes equations physics mode shown below uses the temperature variable **T** from the heat transfer mode in the source term for the y-direction.

Moreover, here below we can see how to make a three way multiphysics coupling. The convective velocities **u** and **v** are coupled from the Navier-Stokes equations physics mode and at the same time the temperature **T** and its two derivatives **Tx** and **Ty** are simultaneously coupled to the reaction rate source term in the convection and diffusion mode.

As we have seen, it is very simple to set up multiphysics models in FEATool. This is made possible by the expression parsing functionality that allows you to enter and use complex expressions of dependent variables (for example **u**, **v**, **T**, **c**), their first derivatives (by just appending x or y to the variable names like **Tx** and **Ty**), the space dimensions **x** and **y**, as well as all common MATLAB expressions and constants like **pi**, **sin**, **cos**, **sqrt**, **^2** etc.

FEATool Multiphysics conveniently features built-in parsing and evaluation of non-constant variables and coefficients This allows users to quickly and easily enter modeling expressions formulated much as one would write on paper, all without having to develop, write, and compile external user defined functions as other more complex simulation software often requires.

The following applies to all physics coefficients such as point constraints, boundary conditions, equation coefficients, initial conditions and postprocessing expressions, as well as command line functions such as evalexpr, intsubd, intbdr, minmaxsubd, minmaxbdr, and postplot. A coefficient may be composed as a string expression including any number of the various coefficient types listed below, combined with the operators `+-*/^`

, parentheses pairs `()`

, as well as numeric constants and scalars.

The space coordinates `x`

, `y`

, and `z`

for Cartesian coordinate systems, and `r`

and `z`

for axi-symmetric and cylindrical coordinate systems are valid coefficient expressions. (Note that the space coordinate names can be re-assigned from their default definitions when starting a *New Model*).

The y-coordinate is for example used to define a parabolic velocity inlet profile as *u _{inlet}* =

`4*0.3/0.41^2*y*(0.41-y)`

in the flow around a cylinder CFD tutorial example.Solution unknowns or dependent variables can also be used in coefficients, with their corresponding names given by the included physics modes, such as for example `T`

for temperature. (As with the space coordinate names, dependent variable names may also be re-defined from the default names when a new physics mode is added. Also note that adding dependent variables in coefficients often make a problem more non-linear and harder to solve.)

The introductory heat transfer multiphysics model for example shows how the temperature `T`

is coupled to the fluid flow via the source term `alpha*g*rho*(T-Tc)`

, where *alpha*, *g*, *rho*, and *Tc*, are model constants, while the flow velocities `u`

and `v`

in turn are coupled back and driving the temperature field through the convective terms.

By appending a space coordinate postfix to a dependent variable, for example `Tx`

, the derivatives in the corresponding direction can be evaluated (in the example here *Tx* is equivalent to *∂T\∂x*. The capacitance in the spherical capacitor model tutorial is for example calculated by integrating the expression `eps0*epsr*(Vr^2+Vz^2)*pi*r`

, where *V* is the computed potential.

Second order derivatives, as for example `Txy`

equivalent to *∂ ^{2}T\∂x∂y*, can also be evaluated for dependent variables discretized by the second order Lagrange finite element basis functions.

Note that derivatives of dependent variables are evaluated by direct differentiation of the underlying FEM shape function polynomials and some overall accuracy may be lost in this process. For example, a variable discretized with a first order linear basis function, will feature a piecewise constant first derivative on each element, and zero second derivative (even if a global second derivative would exist analytically). For higher order accuracy with derivatives a projection or gradient reconstruction technique is necessary.

The variable name `t`

is reserved for the current time in instationary simulations (evaluates to *0* for stationary problems). For example, the instationary flow around a cylinder example defines a parabolic and time dependent inlet velocity as *u _{inlet}* =

`6*sin(pi*t/8)*(y*(0.41-y))/0.41^2`

.Boundary coefficients may use the external normals by prefixing a space dimension name with the character *n*, as for example `nx`

for the normal in the x-direction. Boundary normals are evaluated and computed as the outward pointing unit vector from the center of each external cell edge or face.

Logical switch expressions such as `T>0`

and `T<=T0`

are also valid syntax and will evaluate to *1* when true and *0* everywhere else. Valid expressions may include the characters `<`

, `>`

, `<=`

, `>=`

, `&`

, and `|`

(corresponding to the *less than*, *greater than*, *less than or equal*, *greater than or equal*, *and*, and *or* operators).

These types of switch expressions can be used to build quite complex expressions and coefficients, for example in the flow over a backwards facing step model the recirculation region is highlighted and visualized with the postprocessing expression `x/hstep*(u<0)*(y<0)`

. This expression effectively isolates the region in *y < 0* where the x-velocity *u* is negative, and overlays it with the scaled coordinate *x/hstep*, resulting in the following plot

Abruptly varying and discontinuous functions like these can sometimes introduce numerical instabilities if used in equation and boundary conditions. From the perspective of the solver it might then be preferable to use a regularized or smoothed Heaviside function, such as

`(abs(w)<1)*(1-(0.5*(1+w+1/pi*sin(pi*w)))) + (w<-1)`

or a smoothed Dirac delta function

`(w>-1 & w<1)*15/16*(1-2*w^2+w^4)/h_grid`

where *w* is a weighting function defining the transition and width of the smoothed region. For example, with *w* = `(sqrt(x^2+y^2)-0.5)/(2*h_grid)`

a circle with radius *0.5* and transition region width *2*h_grid* is defined as shown below.

Smoothed step and delta functions are commonly used in models with moving and immersed boundaries which cannot be resolved by the mesh. Step functions are used to define the discontinuous materials, and delta functions to define surface tension forces and reactions. For example see the multiphase flow models ex_multiphase1.

All of the common built-in mathematical functions and constants in MATLAB, such as `pi`

, `eps`

, `sqrt`

, `sin`

, `cos`

, `log`

, `exp`

, `abs`

etc., may also be used to compose expressions.

If a combination of the above methods is insufficient to implement an expression, or the expression can not be described analytically one can use a custom MATLAB m-file function. This could be the case if one wants to use tabulated or experimental data as input which must be interpolated, or complex coefficients with memory effects such as hysteresis. For example, if a coefficient is described as

`myfun(x,y,t,T)`

FEATool will first evaluate the inner function arguments (in this case *x*, *y*, *t*, and *T*) and then call the function *myfun* with these arguments. The input arguments can be any combination of the valid coefficients described above. This assumes a function file *myfun.m* can be found on any of the MATLAB search paths and directories. The function must return a numeric array with the same size and dimensions as the input arguments. A simplified function m-file could for example look like the following

function [ result ] = myfun( x, y, t, T ) persistent mydata T0 if( isempty(mydata) ) load( 'path_to_mydata_file', 'data_x', 'data_y', 'data_values' ); T0 = 300; end interpolated_data = interp2( data_x, data_y, data_values, x, y ); result = T0 + (1-cos(2*pi*t))*T.^2.*interpolated_data; index_limit = find( result < 0 ); result(index_limit) = 0;

In a custom function one can then use the usual MATLAB script functions and techniques such as *load* (to load external data), *save*, *interp*, *interp2*, *persistent*, *varargin*, *system* etc. (To see information and syntax about a MATLAB command one can enter `help command`

in the command line interface.)

Note that depending on the grid size the function may be called several times per evaluation step for different grid cell blocks (for efficiency reasons the default setting is to assembly block sizes of 50000 grid cells or less per function call).

The coefficient name `h_grid`

is reserved for the mean grid cell size or diameter, and is primarily used in the artificial and convective stabilization techniques.

Internal and local variables may be prefixed with double underscores `__varname`

, and it is therefore not recommended to name variables with underscores to prevent potential name collisions.

- An easy way to test and check an expression for correctness is to simply plot and visualize it in postprocessing mode.
- For two-dimensional surface plots in the FEATool GUI one can simply click on a point in the plot to evaluate the visualized expression in the point.
- When working in the FEATool GUI it is often convenient to use the Model Constants and Expressions dialog box to define and store modeling expressions. The named expressions are then available everywhere in the GUI from equation coefficients to postprocessing expressions.
- The more non-linearities that are introduced the harder it will be for the solver to converge to a solution. Highly non-linear problems typically require more relaxation (which can be achieved by decreasing the
*Non-linear relaxation parameter*in the Solver Settings) and a very good initial guess. One can try to introduce a scalar weighting coefficient to the non-linear coefficients, and step by step increasing it from*0*(no contributions) to*1*(fully non-linear contributions) while using the previous solution as initial guess. Lastly, for stationary problems one can also employ pseudo time stepping with the Backward-Euler scheme to try to use the time dependent solver to reach a steady state.

By using the predefined FEATool physics modes it is possible to easily and quickly implement models which simulate different physical effects such as fluid flow, structural stresses, chemical reactions, and heat transfer. This section describes how the various physics modes are defined, set up and used. The available physics modes are listed in the following table

Physics Mode | Description | Definition Function |
---|---|---|

Poisson Equation | Poisson equation | poisson |

Convection and Diffusion | Mass transport through convection and diffusion | convectiondiffusion |

Conductive Media DC | Electric potential | conductivemediadc |

Electrostatics | Electrostatics | electrostatics |

Magnetostatics | Magnetostatics | magnetostatics |

Heat Transfer | Heat transport through convection and conduction | heattransfer |

Euler-Bernoulli Beam | Euler-Bernoulli beam theory (1D) | eulerbeam |

Plane Stress | Structural mechanics (2D plane stress approximation) | planestress |

Plane Strain | Structural mechanics (2D plane strain approximation) | planestrain |

Axisymmetric Stress-Strain | Structural mechanics (2D cylindrical coordinates) | axistressstrain |

Linear Elasticity | Structural mechanics (3D solid linear elasticity) | linearelasticity |

Darcy's Law | Darcy's law porous media flow | darcyslaw |

Brinkman Equations | Brinkman equations porous media flow | brinkmaneqns |

Navier-Stokes Equations | Incompressible fluid flow | navierstokes |

Swirl Flow | Axisymmtric incompressible fluid flow with swirl | swirlflow |

Euler Equations | Compressible inviscid fluid flow | compressibleeuler |

Custom Equation | User defined equation | customeqn |

The equation parameters and coefficients for each selected physics mode are defined in the corresponding *Equation Settings* dialog boxes described below. In addition to also displaying the PDE equation, it is possible to change or edit the equation definitions with the **edit** equation button, and activate/deactivate physics modes in specific subdomains with the **active** button. Furthermore, the used finite element shape functions can also be selected either from the drop down box or directly entering a space separated list of shape functions for each dependent variable. Physics modes with convective effects allowing for numerical artificial stabilization also feature a button opening the stabilization settings.

The Poisson equation physics mode solves the classic elliptic Poisson equation for the scalar dependent variable \(u\)

\[ d_{ts}\frac{\partial u}{\partial t} + \nabla\cdot(-D\nabla u) = f \]

where \(d_{ts}\) is a time scaling coefficient, \(D\) is a diffusion coefficient, and \(f\) is a scalar source term. In the Poisson *Equation Settings* dialog box shown below these equation coefficients, initial value \(u_0 = u(t=0)\) can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

The physics mode allows for both Dirichlet and Neumann (flux) boundary conditions. Dirichlet conditions prescribe a fixed value \(r\) of the dependent variable \(u = r\) on a boundary segment, while a Neumann condition will prescribe the normal flux to a boundary segment, that is \(-\mathbf{n}\cdot D\nabla u = g\), where \(\mathbf{n}\) is the outward directed normal, and \(g\) therefore represents the value of the inward directed flux. The available boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Dirichlet | Prescribed value, \(u = r\) | r |

Neumann | Prescribed flux, \(-\mathbf{n}\cdot D\nabla u = g\) | g |

The convection and diffusion physics mode models mass transport and reaction of a chemical species \(c\). The governing equation for convection and diffusion reads

\[ d_{ts}\frac{\partial c}{\partial t} + \nabla\cdot(-D\nabla c) = R - \mathbf{u}\cdot\nabla c \]

where \(d_{ts}\) is a time scaling coefficient, \(D\) is a diffusion coefficient, \(R\) is the reaction rate source term, and \(\mathbf{u}\) a vector valued convective velocity field. In the *Equation Settings* dialog box shown below these equation coefficients, initial value \(c_0 = c(t=0)\) can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

For convection and diffusion problems with dominating convective effects it is advisable to use artificial stabilization. Pressing the lower **Artificial Stabilization** button opens the corresponding dialog box which allows adding isotropic artificial diffusion, anisotropic streamline diffusion, and shock capturing stabilization. Turning coefficients are also provided to control the strength of the introduced artificial diffusion.

The convection and diffusion physics mode allows for four different boundary conditions; a prescribed concentration boundary condition, a convective flow (outflow condition), an insulation/symmetry condition which prescribes zero flux (or flow), and a prescribed flux condition. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Concentration | Prescribed concentration, \(c = c_0\) | c_{0} |

Convective flux | Outflow, \(-\mathbf{n}\cdot D\nabla c = 0\) | |

Insulation/symmetry | Zero flux/flow, \(-\mathbf{n}\cdot (D\nabla c + \mathbf{u}c) = 0\) | |

Flux condition | Prescribed flux, \(-\mathbf{n}\cdot (D\nabla c + \mathbf{u}c) = N_0\) | N_{0} |

Electric potential and current can be modeled through the conductive media DC physics mode. To describe the flow of current and electric potential \(V\) an electric field is firstly defined as \(\mathbf{E} = \nabla V\). Furthermore, the current density \(\mathbf{J}\) is related to the electric field by \(\mathbf{J} = \sigma\mathbf{E}\) where \(\sigma\) is the conductivity. By assuming conservation of the current one can pose a continuity equation for the current density, that is \(\nabla\cdot\mathbf{J} = Q\) where \(Q\) is a current source, which after expansion results in the following equation

\[ d_{ts}\frac{\partial V}{\partial t} + \nabla\cdot(-\sigma\nabla V) = Q \]

In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the electric potential \(V_0 = V(t=0)\) can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

The conductive media DC physics mode features two boundary conditions, one prescribing the electric potential at a boundary \(V = V_0\), and the other prescribing the current flow into the boundary. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Electric potential | Prescribed electric potential, \(V = V_0\) | V_{0} |

Current flow | Prescribed current flow (flux), \(-\mathbf{n}\cdot (\sigma\nabla V) = J_0\) | J_{0} |

The electrostatics physics mode is an extension of the conductive media DC mode allowing for polarization vector, *P*, to be taken into account. The electrostatics equation for the electric potential *V* then reads

\[ -\nabla\cdot(\epsilon\nabla V - \mathbf{P}) = \rho \]

where \(\epsilon\) is the permittivity and \(\rho\) the charge density source term.

In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the electric potential \(V_0 = V(t=0)\) can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

The electrostatics physics mode features two boundary conditions, one prescribing the electric potential at a boundary \(V = V_0\), and the other prescribing the current flow into the boundary. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Electric potential | Prescribed electric potential, \(V = V_0\) | V_{0} |

Ground/antisymmetry | Zero electric potential, \(V = 0\) | |

Surface charge | Prescribed current flow (flux), \(-\mathbf{n}\cdot (-\epsilon\nabla V + \mathbf{P}) = \rho_s\) | rho_{s} |

Insulation/symmetry | Zero current flow (flux), \(-\mathbf{n}\cdot (-\epsilon\nabla V + \mathbf{P}) = 0\) |

The magnetostatics physics mode models simplified Maxwell's equations and in two dimensions solves

\[ -\nabla\cdot( \frac{1}{\mu}\nabla A_z - \mathbf{M} ) = J_z \]

for the magnetic vector potential \(A_z\) in the z-direction where \(\epsilon\) is the permeability, \(\mathbf{M}\) the magnetization vector, and \(J_z\) the current density. In three dimensions the equations are simplified to to exclude currents which reduces them to solving for a scalar magnetic potential \(V_m\)

\[ -\nabla\cdot( \mu\nabla V_m - \mu\mathbf{M} ) = 0 \]

In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the magnetic potential can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

The magnetostatics physics mode features four boundary conditions, one prescribing the magnetic potential at a boundary, one for magnetic insulation zero potential, surface current, and magnetic insulation/symmetry. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Magnetic potential | Prescribed magnetic potential, \(A_z/V_m = A_{z,0}/V_{m,0}\) | Az_{0}/Vm_{0} |

Magnetic insulation/antisymmetry | Zero magnetic potential, \(A_z/V_m = 0\) | |

Surface current | Prescribed current flow (flux), \(-\mathbf{n}\cdot (\frac{1}{\mu}\nabla A_z - \mathbf{M}/\mu\nabla V_m - \mu\mathbf{M}) = \mathbf{J}_s\) | J_{x/y/z,s} |

Electric insulation/symmetry | Zero current flow (flux), \(-\mathbf{n}\cdot (\frac{1}{\mu}\nabla A_z - \mathbf{M}/\mu\nabla V_m - \mu\mathbf{M}) = 0\) |

The heat transfer physics mode models heat transport through convection and conduction and heat generation through the following governing equation

\[ \rho C_p\frac{\partial T}{\partial t} + \nabla\cdot(-k\nabla c) = Q - \rho C_p\mathbf{u}\cdot\nabla T \]

where \(\rho\) is the density, \(C_p\) the heat capacity, \(k\) is the thermal conductivity, \(Q\) is the heat source term, and \(\mathbf{u}\) a vector valued convective velocity field. In the *Equation Settings* dialog box show below the equation coefficients, initial value for the temperature \(T_0 = T(t=0)\) can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

For heat transfer problems with dominating convective effects it is advisable to use artificial stabilization. Pressing the lower **Artificial Stabilization** button opens the corresponding dialog box which allows adding isotropic artificial diffusion, anisotropic streamline diffusion, and shock capturing stabilization. Turning coefficients are also provided to control the strength of the introduced artificial diffusion.

The heat transfer physics mode allows for four different boundary conditions; prescribed temperature, convective flow (outflow condition), an insulation/symmetry condition which prescribes zero flux (or flow), and a prescribed flux boundary condition.

The heat flux boundary condition involves several parameters. Firstly, an arbitrary expression for the heat flux may be prescribed with the coefficient **q _{0}**. The second term,

The boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Temperature | Prescribed temperature, \(T = T_0\) | T_{0} |

Convective flux | Outflow, \(-\mathbf{n}\cdot k\nabla T = 0\) | |

Thermal insulation/symmetry | Zero heat flux, \(-\mathbf{n}\cdot (k\nabla T + \rho C_p\mathbf{u}c) = 0\) | |

Heat flux | Prescribed heat flux, \(-\mathbf{n}\cdot (k\nabla T + \rho C_p\mathbf{u}T) = q_0 + h\cdot(T_{inf}-T) + Const\cdot(T_{amb}^4-T^4)\) | q_{0}, h, T_{inf}, Const, T_{amb} |

The Euler-Bernoulli beam physics mode models displacements, stresses and strains in a one-dimensional representations of beams and bars. The Euler-Bernoulli equations for the displacement \(v\) reads

\[ \rho A \frac{\partial^2 v}{\partial t^2} + \frac{\partial^2}{\partial x^2}(EI\frac{\partial^2 v}{\partial x^2}) = q \]

where \(\rho\) is the beam material density, *A* the cross-sectional area, *E* modulus of elasticity, *I* cross section moment of inertia, and *q* any distributed loads. In the Euler beam *Equation Settings* dialog box shown below these equation coefficients, initial value \(v_0 = v(t=0)\) can be specified. The FEM shape function space can also be selected from the drop-down combobox (Hermite C1 shape functions by default).

The boundary conditions for the Euler beam physics mode allows for any combination of prescribing displacements and edge loads. These boundary conditions are summarized in the table below.

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Fixed displacements | Prescribed displacements, \( u = u_0, v = v_0\) | v_{0} |

Edge loads | Prescribed edge loads, \( F_x = f_{x,0}, F_y = f_{y,0} \) | f_{y,0} |

The plane stress physics mode models how structural stresses form in thin structures where the planar component of the stress can be neglected or considered zero. In this case the stress-strain relations can be written as

\[ \left( \begin{array}{c} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{array} \right) = \frac{E}{1-\nu^2} \left(\begin{array}{ccc} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{array} \right) \left( \begin{array}{c} \epsilon_x \\ \epsilon_y \\ \gamma_{xy} \end{array} \right) \]

where \(E\) is the elastic or Young's modulus, and \(\nu\) is the Poisson's ratio of the material. The strains are related to the material displacements ( \(u\), \(v\)) as

\[ \epsilon_x = \frac{\partial u}{\partial x}, \qquad \epsilon_y = \frac{\partial v}{\partial y}, \qquad \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \]

Balance equations for the stresses finally give the resulting governing equation system as

\[ \left\{\begin{aligned} - \frac{\partial\sigma_x}{\partial x} - \frac{\partial\tau_{xy}}{\partial y} = F_x \\ - \frac{\partial\tau_{xy}}{\partial x} - \frac{\partial\sigma_y}{\partial y} = F_y \end{aligned}\right. \]

where \(F_x\) and \(F_y\) are volume (body) forces in the x and y-directions, respectively. In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the displacements can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

Optional stress-strain temperature dependence can also be added, which takes the form

\[ \mathbf{\sigma} = \mathbf{D\epsilon} - \frac{E\alpha T}{1-\nu}[1\ \ 1\ \ 0]^T \]

where \(\alpha\) is the coefficient of thermal expansion and \(T\) either a prescribed temperature field, a dependent variable name from another physics mode that represents the temperature, or a combination such as \(T-T_{ref}\).

The boundary conditions for the plane stress physics mode allows for any combination of prescribing displacements and edge loads. These boundary conditions are summarized in the table below.

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Fixed displacements | Prescribed displacements, \( u = u_0, v = v_0\) | u_{0}, v_{0} |

Edge loads | Prescribed edge loads, \( F_x = f_{x,0}, F_y = f_{y,0} \) | f_{x,0}, f_{y,0} |

Like the plane stress physics mode, the plane strain mode models how structural stresses form in structures but where the z-component of the displacements can be neglected or considered zero. In this case the stress-strain relations can be written as

\[ \left( \begin{array}{c} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{array} \right) = \frac{E}{(1+\nu)(1-2\nu)} \left(\begin{array}{ccc} {1-\nu} & \nu & 0 \\ \nu & {1-\nu} & 0 \\ 0 & 0 & \frac{1-2\nu}{2} \end{array}\right) \left( \begin{array}{c} \epsilon_x \\ \epsilon_y \\ \gamma_{xy} \end{array} \right) \]

where \(E\) is the elastic or Young's modulus, and \(\nu\) is the Poisson's ration of the material. The strains are related to the material displacements ( \(u\), \(v\)) as

\[ \epsilon_x = \frac{\partial u}{\partial x}, \qquad \epsilon_y = \frac{\partial v}{\partial y}, \qquad \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \]

Balance equations for the stresses finally give the resulting governing equation system as

\[ \left\{\begin{aligned} - \frac{\partial\sigma_x}{\partial x} - \frac{\partial\tau_{xy}}{\partial y} = F_x \\ - \frac{\partial\tau_{xy}}{\partial x} - \frac{\partial\sigma_y}{\partial y} = F_y \end{aligned}\right. \]

where \(F_x\) and \(F_y\) are volume (body) forces in the x and y-directions, respectively. In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the displacements can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

Optional stress-strain temperature dependence can also be added, which takes the form

\[ \mathbf{\sigma} = \mathbf{D\epsilon} - \frac{E\alpha T}{(1+\nu)(1-2\nu)}[1\ \ 1\ \ 0]^T \]

where \(\alpha\) is the coefficient of thermal expansion and \(T\) either a prescribed temperature field, a dependent variable name from another physics mode that represents the temperature, or a combination such as \(T-T_{ref}\).

The boundary conditions for the plane strain physics mode allows for any combination of prescribing displacements and edge loads. These boundary conditions are summarized in the table below.

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Fixed displacements | Prescribed displacements, \( u = u_0, v = v_0\) | u_{0}, v_{0} |

Edge loads | Prescribed edge loads, \( F_x = f_{x,0}, F_y = f_{y,0} \) | f_{x,0}, f_{y,0} |

The axisymmetric stress-strain physics mode models how stresses and strains in cylindrical and rotationally symmetric geometries behave. The model equations are reduced from full 3D to a two-dimensional slice. In this case the balance equations for stress-strain (valid for the half plane *r>0*, with *r=0* being the symmetry axis) can be written as

\[ \left\{\begin{aligned} - \frac{\partial\sigma_r}{\partial r} - \frac{\partial\tau_{rz}}{\partial z} - \frac{\sigma_r - \sigma_\theta}{r} = F_r \\ - \frac{\partial\tau_{rz}}{\partial r} - \frac{\partial\sigma_z}{\partial z} - \frac{\tau_{rz}}{r} = F_z \end{aligned}\right. \]

where \(F_r\) and \(F_z\) are volume (body) forces in the r and z-directions, respectively. The balance equations together with the constitutive relations

\[ \left( \begin{array}{c} \sigma_r \\ \sigma_\theta \\ \sigma_z \\ \tau_{rz} \end{array} \right) = \frac{E\nu}{(1+\nu)(1-2\nu)} \left(\begin{array}{cccc} \frac{1-\nu}{\nu} & 1 & 1 & 0 \\ 1 & \frac{1-\nu}{\nu} & 1 & 0 \\ 1 & 1 & \frac{1-\nu}{\nu} & 0 \\ 0 & 0 & 0 & \frac{1-2\nu}{2\nu} \end{array} \right) \left( \begin{array}{c} \epsilon_r \\ \epsilon_\theta \\ \epsilon_z \\ \gamma_{rz} \end{array} \right) \]

uniquely define the equations to solve. Here \(E\) is the elastic or Young's modulus, and \(\nu\) is the Poisson's ratio of the material. The strains are related to the material displacements ( \(u\), \(w\)) as

\[ \epsilon_r = \frac{\partial u}{\partial r}, \qquad \epsilon_\theta = \frac{u}{r}, \qquad \epsilon_z = \frac{\partial w}{\partial z}, \qquad \gamma_{rz} = \frac{\partial u}{\partial z} + \frac{\partial w}{\partial r} \]

In the PDE equation formulations the dependent variable *u* is replaced by *u/r* in order to avoid divisions by zero on the symmetry line (Thus custom postprocessing expressions including *u* should instead use *r*u*).

In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the displacements can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

Optional stress-strain temperature dependence can also be added, which takes the form

\[ \mathbf{\sigma} = \mathbf{D\epsilon} - \frac{E\alpha T}{1-\nu}[1\ \ 1\ 1\ \ 0]^T \]

where \(\alpha\) is the coefficient of thermal expansion and \(T\) either a prescribed temperature field, a dependent variable name from another physics mode that represents the temperature, or a combination such as \(T-T_{ref}\).

The boundary conditions for the plane stress physics mode allows for any combination of prescribing displacements and edge loads. These boundary conditions are summarized in the table below.

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Fixed displacements | Prescribed displacements, \( u = u_0, w = w_0\) | u_{0}, w_{0} |

Edge loads | Prescribed edge loads, \( F_r = f_{r,0}, F_z = f_{z,0} \) | f_{r,0}, f_{z,0} |

The linear elasticity physics mode models how structural stresses form in solid structures where the stress-strain relations can be written as

\[ \left( \begin{array}{c} \sigma_x \\ \sigma_y \\ \sigma_z \\ \tau_{xy} \\ \tau_{yz} \\ \tau_{xz} \end{array} \right) = \frac{E}{(1+\nu)(1-2\nu)} \left(\begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} \end{array} \right) \left( \begin{array}{c} \epsilon_x \\ \epsilon_y \\ \epsilon_z \\ \gamma_{xy} \\ \gamma_{yz} \\ \gamma_{xz} \end{array} \right) \]

where \(E\) is the elastic or Young's modulus, and \(\nu\) is the Poisson's ratio of the material. The strains are related to the material displacements ( \(u\), \(v\), \(w\)) as

\[ \epsilon_x = \frac{\partial u}{\partial x}, \qquad \epsilon_y = \frac{\partial v}{\partial y}, \qquad \epsilon_z = \frac{\partial w}{\partial z}, \qquad \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}, \qquad \gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}, \qquad \gamma_{xz} = \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \]

Balance equations for the stresses finally give the resulting governing equation system as

\[ \left\{\begin{aligned} - \frac{\partial\sigma_x}{\partial x} - \frac{\partial\tau_{xy}}{\partial y} - \frac{\partial\tau_{xz}}{\partial z} = F_x \\ - \frac{\partial\tau_{xy}}{\partial x} - \frac{\partial\sigma_y}{\partial y} - \frac{\partial\tau_{yz}}{\partial z} = F_y \\ - \frac{\partial\tau_{xz}}{\partial x} - \frac{\partial\tau_{yz}}{\partial y} - \frac{\partial\sigma_z}{\partial z} = F_z \end{aligned}\right. \]

where \(F_x\), \(F_y\), and \(F_z\) are volume (body) forces in the x, y, and z-directions, respectively. In the *Equation Settings* dialog box shown below the equation coefficients, initial value for the displacements can be specified. The FEM shape function space can also be selected from the drop-down combobox (1st through 5th order conforming P1/Q1 shape functions), or further specified in the corresponding edit field.

Optional stress-strain temperature dependence can also be added, which takes the form

\[ \mathbf{\sigma} = \mathbf{D\epsilon} - \frac{E\alpha T}{1-2\nu}[1\ \ 1\ \ 1\ \ 0\ \ 0\ \ 0]^T \]

The boundary conditions for the linear elasticity physics mode allows for any combination of prescribing displacements and edge loads. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Fixed displacements | Prescribed displacements, \( u = u_0, v = v_0, w = w_0\) | u_{0}, v_{0} , w_{0} |

Edge loads | Prescribed edge loads, \( F_x = f_{x,0}, F_y = f_{y,0}, F_z = f_{z,0} \) | f_{x,0}, f_{y,0}, f_{z,0} |

Darcy's law models the pressure *p* in porous media flow through the relation

\[ d_{ts}\frac{\partial p}{\partial t} + \nabla\cdot(-\frac{\kappa}{\eta}\nabla p) = F \]

where \(\kappa\) represents permeability, \(\eta\) viscosity, *F* sources and sinks, and \(d_{ts}\) is a time scaling coefficient. In the Darcy's Law *Equation Settings* dialog box shown below the equation coefficients, initial values for the pressure can be specified. The FEM shape function spaces can also be selected from the drop-down combobox, or further specified in the corresponding edit field.

The Darcy's law physics mode allows prescription of three different boundary conditions. The prescribed pressure, flux, and insulation boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Pressure | Prescribed pressure, \(\mathbf{p} = p_0\) | p_{0} |

Insulation/symmetry | Zero flux/flow, \(-\mathbf{n}\cdot(-\frac{\kappa}{\eta}\nabla p) = 0\) | |

Flux | Prescribed flux/flow, \(-\mathbf{n}\cdot(-\frac{\kappa}{\eta}\nabla p) = N_0\) | N_{0} |

The Brinkman equations physics mode models porous media flows and can be seen as a combination of Darcy's law with the Navier-Stokes equations. The Brinkman equations are defined as follows

\[ \left\{\begin{aligned} \rho\frac{\partial\mathbf{u}}{\partial t} - \nabla\cdot(\mu(\nabla\mathbf{u}+\nabla\mathbf{u}^T)) + \frac{\mu}{\kappa}\mathbf{u} + \nabla p &= \mathbf{F} \\ \nabla\cdot\mathbf{u} &= 0 \end{aligned}\right. \]

which is to be solved for the unknown velocity field \(\mathbf{u}\) and pressure \(p\). In these equations \(\rho\) represents the density of the fluid, \(\mu\) the viscosity, and \(\kappa\) permeability, moreover \(\mathbf{F}\) represents body forces acting on the fluid. In the *Equation Settings* dialog box shown below the equation coefficients, initial values for the velocities and pressures can be specified. The FEM shape function spaces can also be selected from the drop-down combobox (P2P1/Q2Q1 and P1P-1/Q2P1 shape functions), or further specified in the corresponding edit field.

The Brinkman equations physics mode allows prescription of several different boundary conditions. Firstly, the no-slip (zero velocity) boundary condition which is appropriate for stationary walls. Moreover, a prescribed velocity condition can be prescribed to both in and outflows as well as moving walls. Prescribed pressure and neutral (zero viscous stress) conditions are both appropriate for outflows. Lastly, symmetry or slip conditions sets the flow to zero in normal direction so as to prevent flow normal to the boundary. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Wall/no-slip | Zero velocity, \(\mathbf{u} = 0\) | |

Inlet/velocity | Prescribed velocity, \(\mathbf{u} = (u_0, v_0, w_0)\) | u_{0}, v_{0}, w_{0} |

Neutral outflow/stress boundary | Zero stress, \(-\mathbf{n}\cdot (-p\mathbf{I} + \mu(\nabla\mathbf{u}+\nabla\mathbf{u}^T)) = 0\) | |

Outflow/pressure | Prescribed pressure, \(p = p_0\) | p_{0} |

Symmetry/slip | Zero normal velocity \(\mathbf{n}\cdot\mathbf{u} = 0\) |

The Navier-Stokes equations physics mode models flows of incompressible fluid flows and which is described by

\[ \left\{\begin{aligned} \rho\left( \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u}\right) - \nabla\cdot(\mu(\nabla\mathbf{u}+\nabla\mathbf{u}^T)) + \nabla p &= \mathbf{F} \\ \nabla\cdot\mathbf{u} &= 0 \end{aligned}\right. \]

which is to be solved for the unknown velocity field \(\mathbf{u}\) and pressure \(p\). In these equations \(\rho\) represents the density of the fluid and \(\mu\) the dynamic viscosity, moreover \(\mathbf{F}\) represents body forces acting on the fluid. In the *Equation Settings* dialog box shown below the equation coefficients, initial values for the velocities and pressures can be specified. The FEM shape function spaces can also be selected from the drop-down combobox (P2P1/Q2Q1 and P1P-1/Q2P1 shape functions), or further specified in the corresponding edit field.

For flow problems with dominating convective effects it is advisable to use artificial stabilization. Pressing the lower **Artificial Stabilization** button opens the corresponding dialog box which allows adding isotropic artificial diffusion, anisotropic streamline diffusion, shock capturing, and pressure stabilization (for first order discretizations). Turning coefficients are also provided to control the strength of the introduced artificial diffusion.

The Navier-Stokes equations physics mode allows prescription of several different boundary conditions. Firstly, the no-slip (zero velocity) boundary condition which is appropriate for stationary walls. Moreover, a prescribed velocity condition can be prescribed to both in and outflows as well as moving walls. Prescribed pressure and neutral (zero viscous stress) conditions are both appropriate for outflows. Lastly, symmetry or slip conditions sets the flow to zero in the normal direction so as to prevent flow normal to the boundary. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Wall/no-slip | Zero velocity, \(\mathbf{u} = 0\) | |

Inlet/velocity | Prescribed velocity, \(\mathbf{u} = (u_0, v_0, w_0)\) | u_{0}, v_{0}, w_{0} |

Neutral outflow/stress boundary | Zero stress, \(-\mathbf{n}\cdot (-p\mathbf{I} + \mu(\nabla\mathbf{u}+\nabla\mathbf{u}^T)) = 0\) | |

Outflow/pressure | Prescribed pressure, \(p = p_0\) | p_{0} |

Symmetry/slip | Zero normal velocity \(\mathbf{n}\cdot\mathbf{u} = 0\) |

The swirl flow physics mode is an extension of Axisymmetric Navier-Stokes equations which allows for non-zero azimuthal velocity (swirl effects). The 2D physics mode allows for a three dimensional velocity field with corresponding momentum equations.

The Euler Equations for inviscid compressible flows is described by

\[ \left\{\begin{aligned} & \frac{\partial\rho}{\partial t} + (\mathbf{u}\cdot\nabla)\rho + \rho(\nabla\cdot\mathbf{u}) = 0 \\ & \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} + \frac{1}{\rho}\nabla p = \frac{1}{\rho}\mathbf{F} \\ & \frac{\partial p}{\partial t} + (\mathbf{u}\cdot\nabla)p + \gamma p(\nabla\cdot\mathbf{u}) = (\gamma-1)(Q-(\mathbf{F}\cdot\mathbf{u})) \end{aligned}\right. \]

which is to be solved for the primitive variables density \(\rho\), velocity field \(\mathbf{u}\) and pressure \(p\). In the *Equation Settings* dialog box shown below the ratio of specific heats \(\gamma\), source terms **F** and **Q**, and initial values for the density, velocities, and pressures can be specified. The FEM shape function spaces can also be selected from the drop-down combobox, or further specified in the corresponding edit field.

As inviscid flow problems to not feature stabilizing viscosity and also may feature discontinous shocks it is neccessary to add artificial stabilization. Pressing the lower **Artificial Stabilization** button opens the corresponding dialog box which allows adding isotropic artificial diffusion, anisotropic streamline diffusion, and shock capturing. Turning coefficients are also provided to control the strength of the introduced artificial diffusion.

The compressible Euler equations physics mode allows prescription of either all the dependent variables for inlets and outlets, symmetry/slip conditions with zero normal velocity, and a neutral/no stress condition appropriate for outlets. These boundary conditions are summarized in the table below

Boundary Condition | Definition | Boundary Coefficient |
---|---|---|

Inlet/outlet | Prescribed variables, \(\rho = \rho_0,\ \mathbf{u} = (u_0, v_0, w_0),\ p = p_0\) | rho_{0}, u_{0}, v_{0}, w_{0}, p_{0} |

Symmetry/slip | Zero normal velocity \(\mathbf{n}\cdot\mathbf{u} = 0\) | |

Neutral/no stress boundary/outlet | Zero stress boundary |

User defined equations can be prescribed by using the custom equation physics mode.

The equation specification can be accessed by either pressing the **edit** button next to the equation description, or the equation description itself. A dialog box will appear showing one edit field for equation and corresponding dependent variable.

Note that the other physics mode equations are defined similarly and can also be edited by pressing the edit button in the corresponding tabs of the equation settings dialog box.

The syntax for equation specifications tries to as close as possible look like how one would write a partial differential equation with pen and paper. If for example the dependent variable is **u** like in the example above, then **u'** corresponds to the time derivative, **ux** the derivative in the x-direction, **ux_x** a second order derivative in the x-direction (to which partial integration will be applied according to the standard finite element derivation of the weak formulation), and **u_t** is the variable u multiplied with the fem test function, thus it will be assembled to the iteration matrix instead of the right hand side. The following table describes the syntax and legal operators

Syntax | Description | Formula |
---|---|---|

u | Dependent variable name | \(u\) |

x | Space dimension name | \(x\) |

ux | Derivative in x-direction, rhs | \(\frac{\partial u}{\partial x}\) |

u_t | Dep. var multiplied with test function | \(u\cdot v\) |

u_x | Derivative of test function | \(u\cdot \frac{\partial v}{\partial x}\) |

ux_t | Derivative in x-direction | \(\frac{\partial u}{\partial x}\cdot v\) |

ux_x | 2nd derivative in x-direction | \(\frac{\partial u}{\partial x}\cdot\frac{\partial v}{\partial x}\) |

uxx_x | 3rd derivative in x-direction | \(\frac{\partial^2 u}{\partial x^2}\cdot\frac{\partial v}{\partial x}\) |

ux_xx | 3rd derivative in x-direction | \(\frac{\partial u}{\partial x}\cdot\frac{\partial^2 v}{\partial x^2}\) |

uxx_xx | 4th derivative in x-direction | \(\frac{\partial^2 u}{\partial x^2}\cdot\frac{\partial^2 v}{\partial x^2}\) |

+ | Addition | |

- | Subtraction | |

* | Multiplication | |

/ | Division | |

sqrt() | Square root | |

^ | Power | |

() | Delimit by enclosing in parentheses |

where *v* is a test function. The equation syntax parser accepts numeric constants and coefficients defined in the **fea.coef** field. Higher dimensions work analogously and the space dimension **x** can be substituted with the others arbitrarily. For a more complicated example look at Black-Scholes Equation in the tutorials section.

For problems with dominating convective effects such as can be found in convection and diffusion, heat transfer problems with convection, and fluid flow problems one can employ artificial stabilization if the grid size is too coarse to allow convergence.

The **Artificial Stabilization** dialog box allows control of isotropic artificial diffusion, anisotropic streamline diffusion, shock-capturing, and pressure stabilization options. Turning coefficients are also provided to control the strength of the introduced artificial diffusion.

Isotropic artificial diffusion adds diffusion of magnitude **delta*h_grid*|u|** in all directions, where **delta** is the tuning coefficient, **h_grid** the local mean diameter of a grid cell, and **|u|** the magnitude of the convective velocity. Streamline diffusion modifies the finite element test function space and only adds a stabilization coefficient of the form **delta*h_grid/|u|** in the direction of the flow so as to minimize changes to the original problem.

Shock capturing and pressure stabilization (PSPG) are appropriate for first order discretizations (all variables use P1 FEM discretizations). Shock capturing allows minimizes unphysical over and undershoots due to convection and PSPG allows one to use otherwise unstable full first order discretizations for incompressible flow problems.