Targets and weighting functions

Introduction

When all geometrical entities are uploaded, the physics and boundary conditions are set, and the design volume is defined, are you able to dictate what ColdStream needs to optimize. This can be defined through targets. ColdStream also gives its users control over the priority of these targets through weighting functions.

Targets

Introduction

To create a design, one or multiple design targets should be set. A target is a goal that the design process of a component should aim to meet. A design problem can have a single or multiple targets. For multiple target problems, these are often not complementary, therefore in practice, a design will mostly incorporate tradeoffs. To better manage those trade-offs, targets can be of one of two types: an objective or a constraint.
The following article goes into more detail about how changing the weighting values impacts the optimized designs.

To better manage those trade-offs, targets can be of one of two types: (1) an objective or (2) a constraint.

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Important

A design objective is a target or goal which is minimized or maximized. It is a target which is aimed to have “as little as possible” or “as much as possible”. An example goal could be maximal cooling efficiency.
A design constraint is a target which has to be satisfied, in terms of “less than X” or “more than Y”. An example could be weight, or the required pumping power being lower than the available one.

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Note

A target should be measurable and should be rigorously defined. A target is set on an entity (region, subregion, boundary). Every target or every combination of targets can lead to a unique output.

Objectives

  1. PowerDissipationMinimization
  2. PressureLossMinimization
  3. RelativeVolumeMinimization
  4. RelativeVolumeMaximization
  5. TemperatureMinimization
  6. TemperatureSpreadMinimization
  7. TemperatureVarianceMinimization
  8. VelocityVarianceMinimization
  9. VelocitySpreadMinimization
  10. ElectricWirePowerMinimization

PowerDissipationMinimization

Name PowerDissipationMinimization
Goal Minimize the Power Dissipation of the flow through the design.
Applicable to Region Fluid
Subregion Not applicable to any subregion types
Boundary Not applicable to any boundary types

Use this goal function if you want to reduce the viscous and pressure losses of the flow through the design. This is of special importance on complex systems, where the fluid flows through several components, each with specific optimal operating conditions that need to be met. Furthermore, if the flow is driven by pumps or fans, minimizing the power dissipation results in a general improvement of efficiency. The power losses in the flow are usually related to recirculation bubbles, turbulence and sudden changes in the flow velocity, so by using this objective, the flow should become more uniform. In other words, the fluid will flow through the design with minimal energy dissipation, thus increasing the mechanical efficiency.

Mathmatical formulation

J=Sin(Φ(p+ρU22))Sout(Φ(p+ρU22))VIMSJ = \int_{S_{in}}\left(\Phi \left(p +\frac{\rho U^2}{2} \right)\right) - \int_{S{out}} \left(\Phi\left(p+\frac{\rho U^2}{2}\right)\right)ds-\int_{V}{IMS}

Where:

  • JJ is the value of the objective in WW.
  • Φ\Phi is the volumetric flux in m3s\frac{m^3}{s}.
  • UU is the velocity in m/sm/s.
  • pp is the pressure in PaPa.
  • ρ\rho is the density in kgm3\frac{kg}{m^3}.
  • SinS_{in} and SoutS_{out} are respectively the inlet and outlet surfaces in m2m^2.
  • IMSIMS are the internal power sources, such as fans in Wm3\frac{W}{m^3}.

PressureLossMinimization

Name PressureLossMinimization
Goal Minimize the pressure loss of the flow through the design.
Applicable to Region Not applicable to any region types
Subregion Not applicable to any subregion types
Boundary fixedFlowRateInlet
fanInlet
pumpInlet
fanOutlet
pumpOutlet

Use this goal function if you want to reduce the pressure loss of the flow through the design. The pressure loss is measured as the difference in pressure between two points on the flow. This objective is useful for example when the flow is driven by a fan or pump: a higher pressure drop requires a higher input of energy, thus reducing the efficiency of the system. Furthermore, in cases where the pressure is restricted by structural or operational constraints, high pressure losses may require inlet pressures that are not possible. In other words, the difference between the pressure at the target area and the ambient pressure will be minimized. For the majority of cases these are respectively the inlet and the outlet pressures.

Mathematical formulation

J=SA(p)pambJ = \int_{S_A}(p)-p_{amb}

Where:

  • JJ is the value of the objective in PaPa.
  • pp is pressure distribution at the boundary where the constraint is set in PaPa.
  • pambp_{amb} is the ambient pressure in PaPa.
  • SAS_A is the full surface of the boundary in m2m^2.

RelativeVolumeMinimization

Name RelativeVolumeMinimization
Goal Minimize the volume of the design with respect to the available design region.
Applicable to Region Not applicable to any region types
Subregion Design
Boundary Not applicable to any boundary types

Use this goal function if you want to decrease the quantity of added material. This objective by itself, will generate an empty design, however when combined with other objectives or constraints it can be very useful. If the added material has a higher cost, a higher weight or has generally less attractive specifications than the original one, it is important to use as little of it as possible, while meeting the other objectives and constraints.

Mathematical formulation

J=VAdded  StructuresVDesign  RegionJ = \frac{V_{Added\;Structures}}{V_{Design\;Region}}

Where:

  • JJ is the value of the objective in [0,1][0, 1].
  • VAdded  StructuresV_{Added \;Structures} is the volume of the new design, the added material in m3m^3.
  • VDesign  RegionV_{Design\;Region} is the volume of the design region in m3m^3.

Advised to be used together with

As stated earlier, this objective function will on its own return the trivial solution where no structures are being added. Users are thus advised to use this objective always in conjunction with another objective or constraint.


RelativeVolumeMaximization

Name RelativeVolumeMaximization
Goal Maximize the volume of the design with respect to the available design region.
Applicable to Region Not applicable to any region types
Subregion Design
Boundary Not applicable to any boundary types

Use this goal function if you want to increase the quantity of added material. This objective by itself will generate a design that matches the design region, however when combined with other objectives or constraints it can be very useful. If the added material has a lower cost, a lower weight or is more resistant than the secondary material (the material of the parent region), it is important to use as much of it as possible, while meeting the other objectives and constraints. In other words, if the added material is “better” than the one it is substituting, this objective will substitute as much as possible.

Mathematical formulation

J=VAdded  StructuresVDesign  RegionJ = \frac{V_{Added\;Structures}}{V_{Design\;Region}}

Where:

  • JJ is the value of the objective in [0,1][0, 1].
  • VAdded  StructuresV_{Added \;Structures} is the volume of the new design, the added material in m3m^3.
  • VDesign  RegionV_{Design\;Region} is the volume of the design region in m3m^3.

Advised to be used together with

As stated earlier, this objective function will on its own return the trivial solution where the design region is completely filled with material. Users are thus advised to use this objective always in conjunction with another objective or constraint.


TemperatureMinimization

Name TemperatureMinimization
Goal Lower the temperature of a component or a region.
Applicable to Region Solid
Fluid
Subregion Design
General
Boundary heatedWall
externalWall

Use this goal function if you want to reduce the temperature across a component by focusing on the highest temperatures. Reducing the highest temperatures on a component will results that its mean temperature will also decrease. Furthermore, the location of the highest temperature in a component may also change.

Mathematical formulation

Region/subregion:

J=1VVTJ = \frac{1}{V}\int_{V}T

Boundary:

J=1SSTJ = \frac{1}{S}\int_S T

Where:

  • JJ is the value of the objective in KK.
  • TT is the temperature distribution on a boundary or a volume in KK.
  • SS is the boundary where the objective is calculated in m2m^2.
  • VV is the (sub)region where the objective is calculated in m3m^3.

TemperatureSpreadMinimization

Name TemperatureSpreadMinimization
Goal Bring the temperature spread across the whole component as close as possible to a specified temperature.
Applicable to Region Solid
Fluid
Subregion Design
General
Boundary heatedWall
externalWall

Use this goal function when you want a component to run at a specified temperature. The spread of temperatures occurring in a component will be made as close as possible to the specified temperature. This can be important for thermal stresses, or when the component functions optimally at a specified temperature. In other words, temperatures of the component will be made as uniformly as possible around the specified temperature. Thus, ideally, the component would have the reference temperature everywhere. If you do not have any reference temperature to set, you can use the temperature variance minimization target instead.

Mathematical formulation

Region/subregion:

J=1VV(TTref)2J=\frac{1}{V}\int_V(T-T_{ref})^2

Boundary:

J=1SS(TTref)2J = \frac{1}{S}\int_S(T-T_{ref})^2

Where:

  • JJ is the value of the objective in K2K^2.
  • TT is the temperature distribution on a boundary or a volume in KK.
  • TrefT_{ref} is the reference temperature to be met in KK.
  • SS is the boundary where the objective is calculated in m2m^2.
  • VV is the (sub)region where the objective is calculated in m3m^3.

TemperatureVarianceMinimization

Name TemperatureVarianceMinimization
Goal Reduce the temperature variations across the whole component.
Applicable to Region Solid
Fluid
Subregion Design
General
Boundary heatedWall
externalWall

Use this goal function if you want to reduce the temperature fluctuations across a component and make the temperature spread as uniform as possible. This can be especially important for thermal stresses. In other words the temperature will be made as uniform as possible across the complete component. If you have a specific reference temperature to focus on, you can use the temperature spread minimization target instead.

Mathematical formulation

Region/subregion:

J=1VV(TTmean)2J=\frac{1}{V}\int_V(T-T_{mean})^2

Boundary:

J=1SS(TTmean)2J = \frac{1}{S}\int_S(T-T_{mean})^2