Problem Class

Class

class problem.Compliance_Problem

This is the Compliance problem class.

Parameters:

name (str) – The Compliance shape optimization problem’s name.

constraint(u, lame_mu, lame_lambda, parameters, measure, xsi=0, vm_DG=0, c_k=0)

Compute the value of constraint.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The integrand of the norm Lp of the Von Mises criteria.

Return type:

ufl.Expression

constraint_integrand(u, lame_mu, lame_lambda, parameters)

Compute the integrand of the area constraint.

Returns:

The integrand of the constraint function.

Return type:

ufl.Expression

cost(u=0, p=0, lame_mu=0, lame_lambda=0, measure=0, Parameters=0)

Compute the cost function, defined as :

\[J\left( \Omega \right) = \int_{\Omega} 0.5 \times \left( 2\mu \sigma(u_{h}):\sigma(u_{h}) + \lambda \nabla \cdot \sigma(u_{h}):\nabla \cdot \sigma(u_{h}) \right) \text{ }dx\]

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

Returns:

The integrand of the cost function.

Return type:

ufl.Expression

cost_integrand(u, lame_mu, lame_lambda, Parameters=0)

Compute the integrand of the cost function, defined as :

\[j\left( u \right) = 0.5 \times \left( 2\mu \sigma(u):\sigma(u) + \lambda \nabla \cdot \sigma(u):\nabla \cdot \sigma(u) \right)\]

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

Returns:

The integrand of the cost function.

Return type:

ufl.Expression

dual_operator(u, lame_mu, lame_lambda, parameters, mesh, measure=0, vm_DG=0, c_k=0)

Compute the dual operator.

Parameters:

• u (fem.Function) – The solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• mesh (Mesh) – The mesh of the domaine \(D\).

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

Return type:

float.

shape_derivative_integrand(u, p, lame_mu, lame_lambda, parameters, measure=0)

Compute the following shape derivative integrand:

\[\begin{split}\begin{align} x =& j\left( u_{h} \right) - 2\mu \sigma(u_{h}):\sigma(p_{h}) + \lambda \nabla \cdot \sigma(u_{h}):\nabla \cdot \sigma(p_{h}) \\ = & - 0.5 \times \left( 2\mu \sigma(u_{h}):\sigma(p_{h}) + \lambda \nabla \cdot \sigma(u_{h}):\nabla \cdot \sigma(p_{h}) \right) \end{align}\end{split}\]

Remark:

For the special case of the compliance minimization, we have \(\\u_{h} = \\p_{h}\)

param fem.Function u:

The solution of the primal problem \(\\u_{h}\).

returns:

The shape derivative of the cost function minus linear elasticity.

rtype:

ufl.Expression

shape_derivative_integrand_constraint(u, p, lame_mu, lame_lambda, parameters, measure=0, vm_DG=0, c_k=0)

Compute the shape derivative integrand of the area constraint.

Returns:

The integrand of the constraint function.

Return type:

ufl.Expression

class problem.VMLp_Problem

This is the Von Mises Lp norm problem class.

Parameters:

name (str) – The Von Mises Lp norm shape optimization problem’s name.

constraint(u, lame_mu, lame_lambda, parameters, measure, xsi=0, vm_DG=0, c_k=0)

Compute the value of constraint.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The integrand of the norm Lp of the Von Mises criteria.

Return type:

ufl.Expression

constraint_integrand(u, lame_mu, lame_lambda, parameters)

Compute the integrand of the area constraint.

Returns:

The integrand of the constraint function.

Return type:

ufl.Expression

cost(u=0, p=0, lame_mu=0, lame_lambda=0, measure=0, Parameters=0)

Compute the cost function, defined as :

\[\begin{split}J\left( \Omega \right) = & \left[ \int_{\Omega} \left[\frac{\sigma_{VM}\left(u_{h}\right)}{\overline{\sigma_{VM}}}\right]^{p}\text{ }dx \right]^{\frac{1}{p}} \\\end{split}\]

with \(\sigma_{VM}\left(u_{h}\right)\) compute with the function mecanics_tool.von_mises() and \(\overline{\sigma_{VM}}\) a constant attribut of Parameters.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The integrand of the cost function.

Return type:

ufl.Expression

cost_integrand(u, lame_mu, lame_lambda, Parameters)

Compute the integrand of the cost function given by the following formula:

\[x = \left[\frac{\sigma_{VM}\left(u_{h}\right)}{\overline{\sigma_{VM}}}\right]^{p}\]

with \(\sigma_{VM}\left(u_{h}\right)\) compute with the function mecanics_tool.von_mises() and \(\overline{\sigma_{VM}}\) a constant attribut of Parameters.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

Returns:

The integrand of the norm Lp of the Von Mises criteria.

Return type:

ufl.Expression

dual_operator(u, lame_mu, lame_lambda, parameters, mesh, measure=0, vm_DG=0, c_k=0)

Compute the dual operator.

Parameters:

• u (fem.Function) – The solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• mesh (Mesh) – The mesh of the domaine \(D\).

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The dual linear form.

Return type:

ufl.Expression

shape_derivative_integrand(u, p, lame_mu, lame_lambda, parameters, measure=0)

Compute the following shape derivative integrand:

\[\begin{split}\begin{align} x = & \frac{1}{p} \left[ \int_{\Omega} \left[\frac{\sigma_{VM}\left(u_{h}\right)}{\overline{\sigma_{VM}}}\right]^{p} \text{ }dx \right]^{\frac{1}{p}-1} \\ & \left[ \left( \frac{\sigma_{VM}\left(u_{h}\right)}{\overline{\sigma_{VM}}}\right)^{p}- 2\mu \sigma(u_{h}):\sigma(p_{h}) + \lambda \nabla \cdot \sigma(p_{h}):\nabla \cdot \sigma(u_{h}) \right] \end{align}\end{split}\]

Parameters:

• u (fem.Function) – The solution of the primal problem \(\\u_{h}\).

• p (fem.Function) – The solution of the dual problem \(\\p_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The shape derivative of the cost function minus linear elasticity.

Return type:

ufl.Expression

shape_derivative_integrand_constraint(u, p, lame_mu, lame_lambda, parameters, measure=0, vm_DG=0, c_k=0)

Compute the shape derivative integrand of the area constraint.

Returns:

The integrand of the constraint function.

Return type:

ufl.Expression

class problem.AreaProblem

This is the Area problem class.

Parameters:

name (str) – The Area shape optimization problem’s name.

constraint(u, lame_mu, lame_lambda, parameters, measure, xsi=0, vm_DG=0, c_k=1)

Compute the value of constraint.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The integrand of the norm Lp of the Von Mises criteria.

Return type:

ufl.Expression

constraint_integrand(u, lame_mu, lame_lambda, parameters)

Compute the integrand of the Lp norm of the Von Mises constraint, defined as:

\[x = \left[\frac{\sigma_{VM}\left(u_{h}\right)}{\overline{\sigma_{VM}}}\right]^{p}\]

with \(\sigma_{VM}\left(u_{h}\right)\) compute with the function mecanics_tool.von_mises() and \(\overline{\sigma_{VM}}\) a constant attribut of Parameters.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

Returns:

The integrand of the constraint fonction.

Return type:

ufl.Expression

cost(u=0, p=0, lame_mu=0, lame_lambda=0, measure=0, Parameters=0)

Compute the cost function, defined as :

\[J\left( \Omega \right) = \int_{\Omega} \text{ }dx\]

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The the cost value.

Return type:

ufl.Expression

cost_integrand(u, lame_mu, lame_lambda, Parameters)

Compute the integrand of the cost function.

Parameters:

• u (fem.Function) – The displacement field, solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

Returns:

The integrand of the norm Lp of the Von Mises criteria.

Return type:

ufl.Expression

dual_operator(u, lame_mu, lame_lambda, parameters, mesh, measure=0, vm_DG=0, c_k=0)

Compute the dual operator.

Parameters:

• u (fem.Function) – The solution of the primal problem \(\\u_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• mesh (Mesh) – The mesh of the domaine \(D\).

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The dual linear form.

Return type:

ufl.Expression

shape_derivative_integrand(u, p, lame_mu, lame_lambda, parameters, measure=0)

Compute the shape derivative integrand of the area constraint.

Parameters:

• u (fem.Function) – The solution of the primal problem \(\\u_{h}\).

• p (fem.Function) – The solution of the dual problem \(\\p_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The integrand of the cost function.

Return type:

ufl.Expression

shape_derivative_integrand_constraint(u, p, lame_mu, lame_lambda, parameters, measure=0, vm_DG=0, c_k=1)

Compute the shape derivative integrand of the constraint.

Parameters:

• u (fem.Function) – The solution of the primal problem \(\\u_{h}\).

• p (fem.Function) – The solution of the dual problem \(\\p_{h}\).

• lame_mu (float) – The \(\mu\) Lame coefficient.

• lame_lambda (float) – The \(\lambda\) Lame coefficient.

• parameters (Parameter) – The parameter object.

• measure (Measure) – The measure on the domain \(\Omega\).

Returns:

The integrand of the constraint function.

Return type:

ufl.Expression