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.

❗️
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.

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

Objectives

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 $\frac{W}{m^3}$ .

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 $m^2$ .

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.

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.

MaximumTemperatureMinimization

Name	MaximumTemperatureMinimization
Goal	Lower the maximum 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 maximum temperature of a component. 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

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 $m^3$ .

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 $m^3$ .

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

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 $m^3$ .

Advised to be used together with

This objective function's sole purpose is to reduce the temperature spread as much as possible. However, this minimal temperature spread may lay at incredibly high temperatures. It is thus advised to use this objective in combination with the a temperatureMinimization objective or maximumTemperature constraint.

VelocityVarianceMinimization

Name	VelocityVarianceMinimization
Goal	Reduce the velocity variations across the whole boundary.
Applicable to	Region	Not applicable to any region
	Subregion	Not applicable to any subregions
	Boundary	pressureInlet pressureOutlet

Use this goal function if you want to reduce the velocity fluctuations across a boundary and make the velocity spread as uniform as possible.

Mathematical formulation

J=\frac{1}{S}\int_S(U-U_{mean})\cdot(U-U_{mean})

Where:

$J$ is the value of the objective in $\frac{m^2}{s^2}$ .

$U$ is the velocity distribution on a boundary or a volume in $\frac{m}{s}$ .

$U_{mean}$ is the mean velocity in the boundary in $\frac{m}{s}$ .

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

Advised to be used together with

This objective function's sole purpose is to reduce the velocity spread as much as possible at the desired boundary patch. However, this minimal velocity spread may be achieved by reducing the fluid velocity to almost 0 m/s. It is thus advised to use this objective in combination with the a powerDissipationMinimization or pressureLossMinimization objective (or their equivalent constraints).

VelocitySpreadMinimization

Name	VelocitySpreadMinimization
Goal	Bring the velocity across the entire entity as close as possible to the specified target velocity.
Applicable to	Region	Not applicable to any region
	Subregion	General subregions of a fluid body
	Boundary	pressureOutlet

Use this goal function if you want to get the velocity on an entity as close as possible to a desired target velocity. This target function is ideal if you have multiple outlets and are not looking for a uniform flow distribution.

Mathematical formulation

J=\frac{1}{S}\int_S(U-U_{target})\cdot(U-U_{target})

Where:

$J$ is the value of the objective in $\frac{m^2}{s^2}$ .

$U$ is the velocity distribution on the entity in $\frac{m}{s}$ .

$U_{target}$ is the target velocity on the entity in $\frac{m}{s}$ .

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

Heat flux maximization/minimization

The heat flux measures the flow of energy per unit area per unit time, [W/m²]. The heat flux can be mathematically formulated using Fourier’s law:

$\overrightarrow{q} = - k ∇T$

Note that the heat flux is a vector quantity, so it has both a direction and a magnitude. The heat flux objective is determined by integrating the heat flux over a surface, to obtain the heat transfer rate

Boundary:

$\dot{Q} = \int_s - k ∇T ∙ \overrightarrow{n} = \int_s q ∙ \overrightarrow{n}$

The heat transfer rate is a scalar quantity, since it represents the total amount of heat passing through the surface.
Note that in ColdStream, the sign of the heat transfer rate still contains an important meaning:

Positive values of Q indicate heat entering the case setup through the boundary.
Negative values of Q indicate heat exiting the case setup through the boundary.

Interpretation & Usage of the Heat flux minimization/maximization objective

Usage of heat flux objectives	Heat enters the case setup through the boundary	Heat exits the case setup through the boundary
You wish to maximize the heat transfer rate passing through the boundary	Heat flux maximization	Heat flux minimization
You wish to minimize the heat transfer rate passing through the boundary	Heat flux minimization	Heat flux maximization

Examples:

Heat enters the case setup through a fixed temperature wall (at high temperature). User would like to cool the component containing the fixed temperature wall as much as possible, by extracting as much heat as
possible. Use a heat flux maximization objective on the fixed temperature wall.
Heat enters the case setup through a heated wall. A coolant circulates through the case setup, and extracts heat. An external wall is defined on a solid surrounding the case setup, through which heat can also escape. User would like to minimize the heat loss to the surroundings or extract as much heat as possible using the coolant. Use a heat flux maximization objective on the external wall.
A region generates heat and is in contact with multiple fixed temperature and external walls (at low temperature), through which the heat exits. User would like to extract as much heat as possible through one specific such wall. Use a heat flux minimization objective on that specific wall.
Heat enters a case setup through a fixed temperature wall. You would like to insulate your case setup from this source of heat. Use a heat flux minimization objective on the fixed temperature

HeatFluxMaximization

Name	HeatFkuxMaximization
Goal	Maximize the volume of the design with respect to the available design region.
Applicable to	Region	Not applicable to any region types
	Subregion	Not applicable to any region types
	Boundary	externalWall fixedTemperatureWall

HeatFluxMinimization

Name	HeatFkuxMinimization
Goal	Maximize the volume of the design with respect to the available design region.
Applicable to	Region	Not applicable to any region types
	Subregion	Not applicable to any region types
	Boundary	externalWall fixedTemperatureWall

Constraints

PowerDissipation

Name	PowerDissipation
Goal	The power dissipation of the flow to be within a certain limit.
Applicable to	Region	Fluid
	Subregion	Not applicable to any subregion types
	Boundary	Not applicable to any boundary types
Target input parameter	Maximal power dissipation $P_{max}$ in $[W]$ : The power dissipation measured has to be lower than this value.

The power dissipated by the flow or equivalently, the required pumping power, has to be lower than the specified value. This constraint places a heavy emphasis on pressure drop. Especially when the inlet flow rate is fixed, this effectively means keeping the pressure drop within a certain limit. Typically, this will result in very sleek or streamlined designs, with smooth transitions in order to avoid recirculation zones, wakes or pressure jumps. Additionally, friction due to boundaries/walls will be reduced.

Mathematical formulation

C=\int_AU\cdot n\left(p+\frac{\rho U^2}{2}\right)+U\cdot \tau dA \le P_{max}

Where:

$C$ is the value of the constraint in $W$ .

$P_{max}$ is the maximal power dissipation in $W$ .

$U$ is the velocity in $\frac{m}{s}$ .

$p$ is the pressure in $Pa$ .

$n$ is the unit vector normal to the flow direction.

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

$\tau$ is the shear stress in $Pa$ .

PressureLoss

Name	PressureLoss
Goal	The static pressure drop on the flow has to be within a certain limit.
Applicable to	Region	Not applicable to any region types
	Subregion	Not applicable to any subregion types
	Boundary	fixedFlowRateInlet pumpInlet fanInlet
Target input parameter	Maximal static pressure at the boundary $p_{max}$ in $[Pa]$ : The pressure loss measured has to be lower than this value.

The static pressure drop in the flow through the design will be equal to or smaller than the prescribed value. This means that a maximal pressure is set for a boundary with respect to the environment pressure. For example, a pressure loss constraint on the inlet, would assume that the outlet is at ambient pressure. This will usually result in a streamlined design. Meaning, limited flow recirculation, smooth flow direction changes and low turbulence generation.

Mathematical formulation

C= \int_A(p)dA - p_{amb}\le p_{max}

Where:

$C$ is the value of the constraint in $Pa$ .

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

$p_{max}$ is the maximal value of the pressure drop in $Pa$ .

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

MinimumRelativeVolume

Name	MinimumRelativeVolume
Goal	The quantity of added material has to be above a prescribed value.
Applicable to	Region	Not applicable to any region types
	Subregion	Design
	Boundary	Not applicable to any boundary types
Target input parameter	$mRV$ (Minimal relative volume): the minimal relative quantity of volume that has to be added during the design procedure. The value should be in the interval $[0,1]$ with 0 meaning no material added and 1 meaning full conversion of the design region.

The quantity of material added will be above a given value that is set relatively to the design region volume (e.g. a value of 0.4 means that at least 40% of the design region has to be filled with the specified design material). Practically, this constraint sets a minimal quantity of material that has to be used to achieve the goal and meet any other constraints. This can be useful in the case of weight constraints and cost constraints, if the added material is cheaper and lighter than the parent region material.

Mathematical formulation

C=\frac{V_{Added\;Structures}}{V_{Design\;Region}} \ge mRV \in [0,1]

Where:

$C$ is the value of the constraint 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 $m^3$ .

$mRV$ is the minimal relative volume required in $[0,1]$ .

MaximumRelativeVolume

Name	MaximumRelativeVolume
Goal	The quantity of added material has to be below a prescribed value.
Applicable to	Region	Not applicable to any region types
	Subregion	Design
	Boundary	Not applicable to any boundary types
Target input parameter	$MRV$ (Maximal relative volume): the maximal relative quantity of volume added during the design procedure. The value should be in the interval $[0,1]$ with 0 meaning no material added and 1 meaning full conversion of the design region.

The quantity of material added will be below a given value that is set relatively to the design region volume. (e.g. a value of 0.4 means that no more than 40% of the parent region can be replaced with the specified design material). Practically, this constraint sets a maximal quantity of material that can be used to achieve the goal and meet any other constraints. This can be useful in the case of weight constraints and cost constraints, if the added material is more expensive than the parent region material.

Mathematical formulation

C = \frac{V_{Added\;Structures}}{V_{Design\;Region}} \le MRV \in [0,1]

Where:

$C$ is the value of the constraint 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 $m^3$ .

$MRV$ is the maximal relative volume required in $[0,1]$ .

MinimumTemperature

Name	MinimumTemperature
Goal	The minimum temperature on a component has to be above a given value.
Applicable to	Region	Solid Fluid
	Subregion	Design General
	Boundary	HeatedWall ExternalWall
Target input parameter	$T_{min}$ in $[K]$ : The minimum allowable temperature on the component.

Use this constraint when you want a component minimum temperature to be above a given value.

Mathematical formulation

Region/subregion:

C=T\ge T_{min}

Where:

$C$ is the value of the constraint in $K$ .

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

$T_{min}$ is the minimal allowable temperature on the boundary or volume in $K$ .

MeanTemperature

Name	MeanTemperature
Goal	The mean temperature on a component has to be within a certain limit.
Applicable to	Region	Solid Fluid
	Subregion	Design General
	Boundary	HeatedWall ExternalWall
Target input parameter	$MaxT_{mean}$ in $[K]$ : The maximum allowable mean $T$ on the component.

Use this constraint when you want a component mean temperature to be below a given value. The temperature spread across the component will not be taken into account, thus uniformity is not guaranteed.

Mathematical formulation

Region/subregion:

C=\int_V\frac TV dV \le MaxT_{mean}

Boundary:

C=\int_A\frac TA dA \le MaxT_{mean}

Where:

$C$ is the value of the constraint in $K$ .

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

$V$ is the volume of the region in $m^3$ .

$A$ is the area of the boundary in $m^2$ .

$MaxT_{mean}$ is the maximal allowed mean temerature in $K$ .

MaximumTemperature

Name	MaximumTemperature
Goal	The maximum temperature on a component has to be below a given value.
Applicable to	Region	Solid Fluid
	Subregion	Design General
	Boundary	HeatedWall ExternalWall
Target input parameter	$T_{max}$ in $[K]$ : The maximum allowable temperature on the component.

Use this constraint when you want a component maximum temperature to be under a given value. This constraint is of special importance when the maximum temperature can be a limiting factor for the correct and safe operation of the component. Examples of where this constraint is necessary to be included:

a component that can be touched by a person,
a component made of a material that suffers quality variations above a given temperature,
a component which operational envelope is temperature restricted.

Mathematical formulation

Region/subregion:

C=T\le T_{max}

Where:

$C$ is the value of the constraint in $K$ .

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

$T_{max}$ is the maximal allowable temperature on the boundary or volume in $K$ .

TemperatureVariance

Name	TemperatureVariance
Goal	The temperature variance on a component has to be within a certain limit.
Applicable to	Region	Solid Fluid
	Subregion	Design General
	Boundary	HeatedWall ExternalWall
Target input parameter	$\sigma^2_{T,max}$ in $[K^2]$ : The maximum variance allowable.

Use this constraint when you want the temperature differences across a component to be within a certain range. Meaning that the difference between the temperature at any given point and the mean temperature is limited. This constraint is of special importance to keep the temperature induced stresses on a component within allowable values.

Mathematical formulation

Region/subregion:

C=\frac1V\int_V(T-T_{mean})^2dV\le\sigma^2_{T,max}

Boundary:

C=\frac1S\int_S(T-T_{mean})^2dS\le\sigma^2_{T,max}

Where:

$C$ is the value of the constraint in $K^2$ .

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

$T_{mean}$ is the mean temperature on the boundary or volume in $K$ .

$\sigma^2_{T,max}$ is the maximal allowable temperature variance on the boundary or volume in $K^2$ .

$V$ is the volume of the region in $m^3$ .

$S$ is the area of the boundary in $m^2$ .

Implementation of the constraint on ColdStream

In theory, ColdStream will penalize everything outside of the desired interval width equally. However, the variance constraint needs to be relaxed for two reasons:

To lower the risk of reaching a local optimum. By relaxing the constraints, ColdStream is far more likely to reach a global optimum.
To lower the risk of non-convergence.

This relaxation means that near the boundaries of the desired interval, there are areas where ColdStream does not fully penalize any violations of the constraint. This means that some tolerance will be expected when using this constraint.

	Perfect constraint	Real constraint
Interval width	Exact	Relaxed
Risk local optimum	High	Low
Risk non-convergence	High	Low

How to interprete the constraint

The input parameter for this constraint is the maximum allowed variance. In order to explain how this input value influences the desired temperature interval, we need to go back to the Gaussian curve. We know that 6 times the standard deviation (which equals the square root of the variance) accounts for 99.7% of the full data. This means that six times the square root of the input value is equal to the desired temperature interval.

For example, if you want to have a $5K$ temperature difference on your target surface. Inserting this into what was explained above will give you: $6\cdot\sigma=5𝐾$ . Reformulating this equation returns the input value for ColdStream: $\sigma^2=\left(\frac56\right)^2=0.6944K^2$ .

❗️Important

Constraint: temperature variance: $\sigma^2=x[K^2]$

$\Rarr \Delta T_{99.7\%}=6\sqrt{x}\;[K]$

VelocityVariance

Name	VelocityVariance
Goal	The velocity variance on a boundary has to be within a certain limit.
Applicable to	Region	Fluid
	Subregion	Not applicable to any subregion types
	Boundary	PressureInlet PressureOutlet
Target input parameter	$\sigma^2_{U,max}$ in $\left[\frac{m^2}{s^2}\right]$ : The maximum variance allowable.

Use this constraint when you want the velocity differences across a boundary to be within a certain value. Meaning, the difference between the velocity at any given point and the mean velocity is limited.

Mathematical formulation

Region/subregion:

C = \frac1S\int_S(U-U_{mean})\cdot(U-U_{mean})dS\le\sigma^2_{U,max}

Where:

$C$ is the value of the constraint in $\frac{m^2}{s^2}$ .

$U$ is the velocity distribution on a boundary in $\frac ms$ .

$U_{mean}$ is the mean velocity on the boundary in $\frac ms$ .

$\sigma^2_{U,max}$ is the maximal allowable velocity variance on the boundary in $\frac{m^2}{s^2}$ .

$S$ is the area of the boundary in $m^2$ .

Implementation of the constraint on ColdStream

In theory, ColdStream will penalize everything outside of the desired interval width equally. However, the variance constraint needs to be relaxed for two reasons:

To lower the risk of reaching a local optimum. By relaxing the constraints, ColdStream is far more likely to reach a global optimum.
To lower the risk of non-convergence.

This relaxation means that near the edges of the desired interval, there are areas where ColdStream does not fully penalize any violations of the constraint. This means that some tolerance will be expected when using this constraint.

	Perfect constraint	Real constraint
Interval width	Exact	Relaxed
Risk local optimum	High	Low
Risk non-convergence	High	Low

How to interpret the constraint

The input parameter for this constraint is the maximum allowed variance. In order to explain how this input value influences the desired velocity interval, we need to go back to the Gaussian curve. We know that 6 times the standard deviation (which equals the square root of the variance) accounts for 99.7% of the full data. This means that six times the square root of the input value is equal to the desired velocity interval.

For example, if you want to have a $5 \frac ms$ velocity difference on your target surface. Inserting this into what was explained above will give you: $6 \cdot \sigma = 5\frac ms$ . Reformulating this equation returns the input value for ColdStream: $\sigma^2=(\frac56)^2 = 0.6944 \frac{m^2}{s^2}$

❗️important

Constraint: velocity variance: $\sigma^2=x[\frac{m^2}{s^2}]$

$\Rarr \Delta U_{99.7\%}=6\sqrt{x}\;[\frac ms]$

Weighting Functions

Introduction

When configuring a design case in ColdStream, multiple targets—encompassing both objectives and constraints—can be assigned. In multi-objective optimizations, the relative importance of each objective is controlled via weighting factors. Constraints, however, must be strictly satisfied whenever physically possible, and their satisfaction always takes priority over minimizing or maximizing objectives.

A weighting value (ranging from 0 to 1) specifies the priority of an objective. A value of 1 assigns the highest priority, while a value approaching 0 assigns the lowest priority.

📘
Note
Multiple objectives can share the exact same weighting factor. For example, assigning a weight of 1 to two separate objectives indicates that both are equally important and hold the highest priority.

Pareto front

Multi-objective optimizations inherently involve conflicting targets. The optimal solution is not a single, unique design, but rather a set of compromised solutions known as the Pareto optimal solution set. The specific final design selected from this set is determined by:

Adjusting the weighting factors of the defined objectives.
Imposing strict physical constraints.

The Pareto front represents the mathematical boundary of optimal solutions where improving one target strictly necessitates the deterioration of at least one other target. Solutions located above the Pareto front are physically feasible but sub-optimal, whereas solutions located below it are physically infeasible.

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Example
In a thermal management problem minimizing both maximum temperature and pressure drop, reducing the temperature further requires generating narrower channels or denser fin structures. This geometric change unavoidably increases the pressure drop across the fluid domain.

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Note
Two distinct objectives of the same type (e.g., two separate temperature minimization objectives applied to different regions) are treated entirely independently by the solver. Their weighting factors are not aggregated or summed.

Pareto front: 2 Pareto optimal solutions — Example of two optimization results over the Pareto front.

Weighting Factors for Objectives

Assigning a higher weighting factor to an objective increases its relative importance within the optimization hierarchy. However, relying exclusively on weighted objectives makes it difficult to predict the exact design point that will be selected along the Pareto front. It merely establishes a directional preference for the solver.

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Example
In a dual-objective scenario, assigning a higher weight to temperature minimization compared to pressure minimization yields a generative design favoring thermal performance over flow efficiency. Reversing the weights prompts the solver to generate a design favoring a lower pressure drop at the expense of higher maximum temperatures.

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Important
Utilizing only objectives provides directional guidance but limits precise control over the specific Pareto optimal solution returned.

Influence of weighting factors for pure multi-objective optimization — How an initial design can be optimized into different points in the Pareto front.

Weighting Factors for Constraints

Constraints establish strict mathematical boundaries that the solver must satisfy at all times. Because they are mandatory bounding limits rather than directional goals, applying weighting factors to prioritize constraints is irrelevant.

Constraints are highly effective for pinpointing a specific area on the Pareto front, thereby narrowing the set of allowable solutions. However, defining constraints that are overly strict or mathematically conflicting can render an optimization problem entirely infeasible (resulting in an empty Pareto optimal solution set).

To mitigate this risk, executing a standard simulation (a "control simulation") on the baseline geometry prior to the design run is highly recommended. This establishes a clear understanding of the baseline physical performance, allowing for realistic and achievable constraint definition. Support engineers are also available to assist with formulating feasible multi-objective problems.

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Example
If a baseline control simulation indicates a maximum geometry temperature of 60°C and a pressure drop of 20 kPa, imposing a strict maximum temperature constraint of 10°C while simultaneously attempting to minimize pressure drop will render the optimization problem physically infeasible.

Influence of constraint on feasible solutions — How a constraint limits an optimization.

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Important
Constraints provide strict control over the final design outcome by enforcing boundaries, but carry the inherent risk of making the optimization problem completely unsolvable if set beyond physical limits.

References

Jyotsna K. Mandal, Somnath Mukhopadhyay, and Paramartha Dutta. multi-objective Optimization. Springer Singapore, 2018
Xu Han and Jie Liu. Introduction to multi-objective Optimization Design. eng. In: Numerical Simulation-based Design. Singapore: Springer Singapore, 2020, pp. 141-151. ISBN:9789811030895.