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.

	Region		Sub region		Boundary
					Inlet				Outlet			Wall
	Solid	Fluid	Design	General	FixedFlowRateInlet	PressureInlet	FanInlet	PumpInlet	PressureOutlet	FanOutlet	PumpOutlet	HeatedWall	ExternalWall	FixedTemperatureWall	InsulatedWall
Objectives
PowerDissipationMinimization
PressureLossMinimization
RelativeVolumeMinimization
RelativeVolumeMaximization
TemperatureMinimization
TemperatureSpreadMinimization
TemperatureVarianceMinimization
VelocityVarianceMinimization
VelocitySpreadMinimization
ElectricWirePowerMinimization
Constraints
PowerDissipation
PressureLoss
MinimumRelativeVolume
MaximumRelativeVolume
MinimumTemperature
MeanTemperature
MaximumTemperature
TemperatureVariance
VelocityVariance
MaximumElectricWirePower
MaximumHeatingTime

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

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

Where:

• $J$ is the value of the objective in $W$.

• $\Phi$ is the volumetric flux in $\frac{m^3}{s}$.

• $U$ is the velocity in $m/s$.

• $p$ is the pressure in $Pa$.

• $\rho$ is the density in $\frac{kg}{m^3}$.

• $S_{in}$ and $S_{out}$ are respectively the inlet and outlet surfaces in $m^2$.

• $IMS$ are the internal power sources, such as fans in $\frac{W}{m^3}$.

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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

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

Where:

• $J$ is the value of the objective in $Pa$.

• $p$ is pressure distribution at the boundary where the constraint is set in $Pa$.

• $p_{amb}$ is the ambient pressure in $Pa$.

• $S_A$ is the full surface of the boundary in $m^2$.

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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

$$J = \frac{V_{Added\;Structures}}{V_{Design\;Region}}$$

Where:

• $J$ is the value of the objective in $[0, 1]$.

• $V_{Added \;Structures}$ is the volume of the new design, the added material in $m^3$.

• $V_{Design\;Region}$ is the volume of the design region in $m^3$.

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.

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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

$$J = \frac{V_{Added\;Structures}}{V_{Design\;Region}}$$

Where:

• $J$ is the value of the objective in $[0, 1]$.

• $V_{Added \;Structures}$ is the volume of the new design, the added material in $m^3$.

• $V_{Design\;Region}$ is the volume of the design region in $m^3$.

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.

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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:

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

Boundary:

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

Where:

• $J$ is the value of the objective in $K$.

• $T$ is the temperature distribution on a boundary or a volume in $K$.

• $S$ is the boundary where the objective is calculated in $m^2$.

• $V$ is the (sub)region where the objective is calculated in $m^3$.

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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:

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

Boundary:

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

Where:

• $J$ is the value of the objective in $K^2$.

• $T$ is the temperature distribution on a boundary or a volume in $K$.

• $T_{ref}$ is the reference temperature to be met in $K$.

• $S$ is the boundary where the objective is calculated in $m^2$.

• $V$ is the (sub)region where the objective is calculated in $m^3$.

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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:

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

Boundary:

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