gemseo.problems.mdo.propane.propane module#

The propane combustion MDO problem.

The Propane MDO problem can be found in [PAG96] and [TM06]. It represents the chemical equilibrium reached during the combustion of propane in air. Variables are assigned to represent each of the ten combustion products as well as the sum of the products.

The optimization problem is as follows:

\[\begin{split}\begin{aligned} \text{minimize the objective function } & f_2 + f_6 + f_7 + f_9 \\ \text{with respect to the design variables } & x_{1},\,x_{3},\,x_{6},\,x_{7} \\ \text{subject to the general constraints } & f_2(x) \geq 0\\ & f_6(x) \geq 0\\ & f_7(x) \geq 0\\ & f_9(x) \geq 0\\ \text{subject to the bound constraints } & x_{1} \geq 0\\ & x_{3} \geq 0\\ & x_{6} \geq 0\\ & x_{7} \geq 0\\ \end{aligned}\end{split}\]

where the System Discipline consists of computing the following expressions:

\[\begin{split}\begin{aligned} f_2(x) & = & 2x_1 + x_2 + x_4 + x_7 + x_8 + x_9 + 2x_{10} - R, \\ f_6(x) & = & K_6x_2^{1/2}x_4^{1/2} - x_1^{1/2}x_6(p/x_{11})^{1/2}, \\ f_7(x) & = & K_7x_1^{1/2}x_2^{1/2} - x_4^{1/2}x_7(p/x_{11})^{1/2}, \\ f_9(x) & = & K_9x_1x_3^{1/2} - x_4x_9(p/x_{11})^{1/2}. \\ \end{aligned}\end{split}\]

Discipline 1 computes \((x_{2}, x_{4})\) by satisfying the following equations:

\[\begin{split}\begin{aligned} x_1 + x_4 - 3 &=& 0,\\ K_5x_2x_4 - x_1x_5 &=& 0.\\ \end{aligned}\end{split}\]

Discipline 2 computes \((x_2, x_4)\) such that:

\[\begin{split}\begin{aligned} K_8x_1 + x_4x_8(p/x_{11}) &=& 0,\\ K_{10}x_{1}^{2} - x_4^2x_{10}(p/x_{11}) &=& 0.\\ \end{aligned}\end{split}\]

and Discipline 3 computes \((x_5, x_9, x_{11})\) by solving:

\[\begin{split}\begin{aligned} 2x_2 + 2x_5 + x_6 + x_7 - 8&=& 0,\\ 2x_3 + x_9 - 4R &=& 0, \\ x_{11} - \sum_{j=1}^{10} x_j &=& 0. \\ \end{aligned}\end{split}\]
class PropaneComb1[source]#

Bases: Discipline

The first set of equations of the propane combustion.

This discipline is characterized by two coupling equations in functional form.

classmethod compute_y0(x_shared)[source]#

Solve the first coupling equation in functional form.

Parameters:

x_shared (ndarray) -- The shared design variables.

Returns:

The coupling variable y0.

Return type:

ndarray

classmethod compute_y1(x_shared)[source]#

Solve the second coupling equation in functional form.

Parameters:

x_shared -- The shared design variables.

Returns:

The coupling variable y1.

Return type:

float

auto_detect_grammar_files: ClassVar[bool] = True#

Whether to find the grammar files automatically.

cache: BaseCache | None#

The execution and linearization data saved according to the cache type.

execution_statistics: ExecutionStatistics#

The execution statistics of the process.

execution_status: ExecutionStatus#

The execution status of the process.

jac: JacobianData#

The Jacobian matrices of the outputs.

The structure is {output_name: {input_name: jacobian_matrix}}.

name: str#

The name of the process.

class PropaneComb2[source]#

Bases: Discipline

The second set of equations of the propane combustion.

This discipline is characterized by two coupling equations in functional form.

classmethod compute_y2(x_shared)[source]#

Solve the third coupling equation in functional form.

Parameters:

x_shared (ndarray) -- The shared design variables.

Returns:

The coupling variable y2.

Return type:

ndarray

classmethod compute_y3(x_shared)[source]#

Solve the fourth coupling equation in functional form.

Parameters:

x_shared -- The shared design variables.

Returns:

The coupling variable y3.

auto_detect_grammar_files: ClassVar[bool] = True#

Whether to find the grammar files automatically.

cache: BaseCache | None#

The execution and linearization data saved according to the cache type.

execution_statistics: ExecutionStatistics#

The execution statistics of the process.

execution_status: ExecutionStatus#

The execution status of the process.

jac: JacobianData#

The Jacobian matrices of the outputs.

The structure is {output_name: {input_name: jacobian_matrix}}.

name: str#

The name of the process.

class PropaneComb3[source]#

Bases: Discipline

The third set of equations of the propane combustion.

This discipline is characterized by three coupling equations in functional form.

classmethod compute_y4(x_shared)[source]#

Solve the fifth coupling equation in functional form.

Parameters:

x_shared -- The shared design variables.

Returns:

The coupling variable y4.

Return type:

float

classmethod compute_y5(x_shared)[source]#

Solve the sixth coupling equation in functional form.

Parameters:

x_shared -- The shared design variables.

Returns:

The coupling variable y5.

classmethod compute_y6(x_shared)[source]#

Solve the seventh coupling equation in functional form.

Parameters:

x_shared -- The shared design variables.

Returns:

The coupling variable y6.

Return type:

float

auto_detect_grammar_files: ClassVar[bool] = True#

Whether to find the grammar files automatically.

cache: BaseCache | None#

The execution and linearization data saved according to the cache type.

execution_statistics: ExecutionStatistics#

The execution statistics of the process.

execution_status: ExecutionStatus#

The execution status of the process.

jac: JacobianData#

The Jacobian matrices of the outputs.

The structure is {output_name: {input_name: jacobian_matrix}}.

name: str#

The name of the process.

class PropaneReaction[source]#

Bases: Discipline

Propane's objective and constraints discipline.

This discipline's outputs are the objective function and partial terms used in inequality constraints.

Notes

The equations have been decoupled (y_i = y_i(x_shared)). Otherwise, the solvers may find iterates for which discipline analyses are not computable.

classmethod f_2(x_shared, y_1, y_2, y_3)[source]#

Compute the first term of the objective function.

It is also a non-negative constraint at system level.

Parameters:
  • x_shared (ndarray) -- The shared design variables.

  • y_1 (ndarray) -- The first coupling variable.

  • y_2 (ndarray) -- The second coupling variable.

  • y_3 (ndarray) -- The third coupling variable.

Returns:

The first term of the objective function.

Return type:

ndarray

classmethod f_6(x_shared, y_1, y_3)[source]#

Compute the second term of the objective function.

It is also a non-negative constraint at system level.

Parameters:
  • x_shared -- The shared design variables.

  • y_1 -- The first coupling variable.

  • y_3 -- The third coupling variable.

Returns:

The second term of the objective function.

Return type:

complex

classmethod f_7(x_shared, y_1, y_3)[source]#

Compute the third term of the objective function.

It is also a non-negative constraint at system level.

Parameters:
  • x_shared -- The shared design variables.

  • y_1 -- The first coupling variable.

  • y_3 -- The third coupling variable.

Returns:

The third term of the objective function.

Return type:

complex

classmethod f_9(x_shared, y_1, y_3)[source]#

Compute the fourth term of the objective function.

It is also a non-negative constraint at system level.

Parameters:
  • x_shared -- The shared design variables.

  • y_1 -- The first coupling variable.

  • y_3 -- The third coupling variable.

Returns:

The fourth term of the objective function.

auto_detect_grammar_files: ClassVar[bool] = True#

Whether to find the grammar files automatically.

cache: BaseCache | None#

The execution and linearization data saved according to the cache type.

execution_statistics: ExecutionStatistics#

The execution statistics of the process.

execution_status: ExecutionStatus#

The execution status of the process.

jac: JacobianData#

The Jacobian matrices of the outputs.

The structure is {output_name: {input_name: jacobian_matrix}}.

name: str#

The name of the process.

get_design_space(to_complex=True)[source]#

Read the design space file.

Parameters:

to_complex (bool) --

Whether the current design point is a complex vector.

By default it is set to True.

Return type:

DesignSpace