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Equation heat

Equazione elettrostatica

FEM

Descrizione

For info about the math of the equation, see the Elmer models manual, section Linear Elasticity.

Usage

1. Add an Elmer solver.
2. Select it in the Tree view.
3. There are several ways to invoke the command:
   • Press the Elasticity equation button.
   • Select the Solve → Mechanical equations → Elasticity equation option from the menu.
4. Change the equation's solver settings or the general solver settings if necessary.

Note: For analyses of nonlinear elasticity you must use the Deformation equation (introduced in 0.21). The Elasticity equation is only for linear elasticity.

Note: If you use more than one CPU core for the solver (introduced in 0.21), you cannot use the default solver settings. However, using just one CPU and the default solver settings is in many cases faster than using several CPUs because the elasticity solver is only fast when DatiLinear Solver Type is set to Direct (the default, described here). For multi-CPU solving one can only use the DatiLinear Direct Method MUMPS. However, MUMPS is not freely available as a direct download.

Solver Settings

For the general solver settings, see the Elmer solver settings.

The elasticity equation provides these special settings:

• DatiCalculate Pangle: If the principal angles should be calculated.
• DatiCalculate Principal: If all stresses should be calculated.
• DatiCalculate Strains: If strains will be calculated. This will also calculate the stresses, even if DatiCalculate Principal or DatiCalculate Stresses is false.
• DatiCalculate Stresses: If stresses should be calculated. Compared to DatiCalculate Principal the Tresca yield criterion and the principal stress will not be calculated.
• DatiConstant Bulk System: See the Elmer manual for more info.
• DatiDisplace Mesh: If mesh can be deformed. This is by default true and must be set to false for eigenfrequency analyses.
• DatiFix Displacement: If displacements or forces are set. thereby DatiModel Lumping is automatically used.
• DatiGeometric Stiffness: Considers the geometric stiffness of the body.
• DatiIncompressible: Computation of incompressible material in connection with viscoelastic Maxwell material and a custom DatiVariable.
• DatiMaxwell Material: Compute the viscoelastic material model.
• DatiModel Lumping: Uses model lumping.
• DatiModel Lumping Filename: File to save the results from the model lumping.
• DatiStability Analysis: If true DatiEigen Analysis becomes a stability analysis (buckling analysis). Otherwise a modal analysis is performed.
• DatiUpdate Transient System: See the Elmer manual for more info.
• DatiVariable: The variable for the elasticity equation. Only change this if DatiIncompressible is set to true in accordance to the Elmer manual.

Eigenvalues:

• DatiEigen Analysis: If an eigen analysis should be performed (calculation of eigenmodes and eigenfrequencies).
• DatiEigen System Complex: Should be true if the eigen system is complex. it must be false for a damped eigen value analyses.
• DatiEigen System Compute Residuals: Computes residuals of the eigen value system.
• DatiEigen System Damped: Set a damped eigen analysis. Can only be used if DatiLinear Solver Type is Iterative.
• DatiEigen System Select: Selection of which eigenvalues are computed. Note that the selection of Largest* cause an infinite run for recent Elmer solver (as of August 2022).
• DatiEigen System Tolerance: Convergence tolerance for iterative eigensystem solve. The default is 100 times the DatiLinear Tolerance.
• DatiEigen System Values: The number of the highest eigenmode that should be calculated.

Equation:

• DatiPlane Stress: Computes solution according to the plane stress situation. Applies only for 2D geometry.

Analysis Feature Information

The elasticity equation takes the following analysis features into account if they are set:

• Fixed boundary condition
• Displacement boundary condition
• Force load
• Initial temperature condition
• Pressure load
• Gravity load
• Spring

Note

• Except for calculations in 2D, for all the above analysis features it is important that they act on a face. Features in 3D set to lines or vertices are not recognized by the Elmer solver.

Eigenmode Analysis

To perform an eigenmode analysis (calculation if the eigenmodes and eigenfrequencies), you need to

1. Set DatiEigen Analysis to true
2. Set DatiDisplace Mesh to false
3. Set DatiEigen System Values to the highest number of eigenmodes you are interested in. The smaller this number the shorter the solver runtime since higher modes can be omitted from computation.
4. Add a fixed boundary condition and set at least one face of the body as fixed.
5. Run the solver.

It is highly recommended to use DatiLinear Solver Type set to Direct (the default) because this is much faster and the results are more accurate.

Buckling Analysis

To perform a buckling analysis, you need to do the same as for an Eigenmode Analysis, and additionally:

• Set DatiStability Analysis to true

Results

The available results depend on the solver settings. If none of the DatiCalculate * settings was set to true, only the displacement is calculated. Otherwise also the corresponding results will be available. If DatiEigen Analysis was set to true all results will be available for every calculated eigenmode.

If DatiEigen Analysis was set to true, the eigenfrequencies are output at the end of the solver log in the solver dialog and also in the document SolverElmerOutput that will be created in the tree view after the solver has finished.

Note: The eigenmode displacement ${\displaystyle {\vec {d}}}$ vector has an arbitrary value since the result is

${\displaystyle \quad {\vec {d}}=c\cdot {\vec {u}}}$

whereas ${\displaystyle {\vec {u}}}$ is the eigenvector and ${\displaystyle c}$ is a complex number.

Equation heat

Equazione elettrostatica

FEM