Targets

In order 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. In order to better manage those trade-offs, targets can be of one of two types: an objective or a constraint.
The following article goes more in detail about how changing the weighting values impact the optimized designs.

In order 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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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.

Mathematical 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 \(m^3/s\).
• \(U\) is the velocity in \(m/s\).
• \(p\) is the pressure in \(Pa\).
• \(\rho\) is the density in \(kg/m^{3}\).
• \(S_{in}\) and \(S_{out}\) are respectively the inlet and outlet surfaces in \(m^2\).
• \(IMS\) are the internal power sources, such as fans in \(W/m^3\).

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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

\[J=\int_{S_A}\left( p \right) - p_{amb} \]

where

• \(J\) is the value of the objective in \(Pa\).
• \(p\) is the 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 \(m^2\).

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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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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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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 \(m^2\).
• \(V\) is the (sub)region where the objective is calculated \(m^3\).

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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: \[J= \frac{1}{V}\int_{V} \left( T-T_{ref}\right)^2 \] Boundary: \[J= \frac{1}{S}\int_{S} \left( T-T_{ref}\right)^2 \]

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 \(K\).
• \(S\) is the boundary where the objective is calculated \(m^2\).
• \(V\) is the (sub)region) where the objective is calculated \(m^3\).

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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: \[J= \frac{1}{V}\int_{V} \left(T-T_{mean}\right)^2 \] Boundary: \[J= \frac{1}{S}\int_{S} \left(T-T_{mean}\right)^2 \]

where

• \(J\) is the value of the objective in \(K^2\)
• \(T\) is the temperature distribution on a boundary or a volume in \(K\).
• \(T_{mean}\) is the mean temperature in the boundary \(K\).
• \(S\) is the boundary where the objective is calculated \(m^2\).
• \(V\) is the (sub)region) where the objective is calculated \(m^3\).

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.

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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

\[J= \frac{1}{S}\int_{S} \left(U-U_{mean}\right) \cdot (U-U_{mean}) \]

where

• \(J\) is the value of the objective in \(m^2/s^2\)
• \(U\) is the velocity distribution on a boundary or a volume in \(m/s\).
• \(U_{mean}\) is the mean velocity in the boundary \(m/s\).
• \(S\) is the boundary where the objective is calculated \(m^2\).

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).

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Use this goal function if you use the design region to determine the circuit of an electric wire which heats the region(s) touching the design region and you want to minimize the required electric power. The voltage supplied to the wire circuit and the electrical resistivity of the wire material have to be specified. The result will be an electric circuit with a single wire with a diameter determined by the 3D printing manufacturing specifications. Voltage or frequency control will not be applied, so the wire length will be varied to reach the minimal electric power. The design region has to have a constant height. The design region material to be specified is the material of the electrical wire.
Use this target in combination with a constraint target specified on another region or boundary, such as minimumTemperature.

Mathematical formulation

\[J= \frac{V^2}{R} = \frac{V^2 \pi D^2}{4 \rho l} \]

where

• \(J\) is the value of the objective, or the electric wire power, in \(W\).
• \(V\) is the supplied voltage in \(V\).
• \(R\) is the electrical resistance in \({\Omega}\).
• \(D\) is the wire diameter, taken from manufacturing specifications in \(m\).
• \(\rho\) is the electrical resistivity of the wire material in \({\Omega m}\).
• \(l\) is the wire length in \(m\).

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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

\[C=\int_{A} U \cdot n (p+\frac{\rho U^{2}}{2}) + U \cdot \tau dA \leq P_{max}\]

where

• \(C\) is the value of the contraint in \(W\).
• \(P_{max}\) is the maximal power dissipation in \(W\).
• \(U\) is the velocity in \(m/s\).
• \(p\) is the pressure in \(Pa\).
• \(n\) is the unit vector normal to the flow direction.
• \(\rho\) is the density in \(kg/m^{3}\).
• \(\tau\) is the shear stress in \(Pa\).

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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

\[C=\int_{A} (p)dA - p_{amb} \leq p_{max}\]

where

• \(C\) is the value of the contraint 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 \(Pa\).

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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 become the new design). 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

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

where

• \(C\) is the value of the contraint 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 \[0, 1\].

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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 become the new design). 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

\[C=\frac{V_{Added\:Stuctures}}{V_{Design\:Region}} \leq MRV \in [0 , 1] \]

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 allowed in \[0, 1\].

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

Mathematical formulation

Region/subregion: \[C= T \geq T_{min} \]

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 \(K\).

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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: \[C= \int_{V} \frac{T}{V} dV \leq Max \, T_{mean} \]

Boundary: \[C=\int_{A} \frac{T}{A} dA \leq Max \, T_{mean} \]

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 \(m^2\).
• \( Max T_{mean}\) is the maximal allowed mean temperature in \(K\).

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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: \[C= T \leq T_{max} \]

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 \(K\).

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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: \[C= \frac{1}{V} \int_{V}(T -T_{mean})^2\:dV \leq \sigma^2_{T,max} \] Boundary: \[C= \frac{1}{S} \int_{S}(T -T_{mean})^2\:dS \leq \sigma^2_{T,max} \]

where

• \(C\) is the value of the constraint in \(K^2\).
• \(T\) is the temperature distribution on a boundary or a 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.

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 5 K temperature difference on your target surface. Inserting this into what was explained above will give you: \(6∗𝜎=5𝐾\). Reformulating this equation returns the input value for ColdStream : \(𝜎^2=(\frac{5}{6})^2=0.69444\space 𝐾^2\)

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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: \[C= \frac{1}{S}\int_{S}(U - U_{mean})\cdot(U - U_{mean})\:dS \leq \sigma^2_{U,max} \]

where

• \(C\) is the value of the constraint in \(m^2/s^2\).
• \(U\) is the velocity distribution on a boundary in \(m/s\).
• \(U_{mean}\) is the mean velocity on the boundary in \(m/s\).
• \(\sigma^2_{U,max}\) is the maximal allowable velocity variance on the boundary in \(m^2/s^2\).
• \(S\) 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.

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 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 m/s velocity difference on your target surface. Inserting this into what was explained above will give you: \(6∗𝜎=5\space\frac{m}{s}\). Reformulating this equation returns the input value for ColdStream : \(𝜎^2=(\frac{5}{6})^2=0.69444\space\frac{m^2}{s^2}\)

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Use this constraint when the design region is used for determining the circuit of an electric wire which heats the region(s) touching the design region, but has a maximum required electric power. The voltage supplied to the wire circuit and the electrical resistivity of the wire material have to be specified. The result will be an electric circuit with a single wire with a diameter determined by the 3D printing manufacturing specifications. Voltage or frequency control will not be applied, so the wire length will be varied to reach a certain electric power. The design region has to have a constant height. The design region material to be specified is the material of the electrical wire.

Mathematical formulation

\[C= \frac{V^2}{R} = \frac{V^2 \pi D^2}{4 \rho l} \leq P_{max}\]

where

• \(C\) is the value of the contraint in \(W\).
• \(P_{max}\) is the maximal electric power in \(W\).
• \(V\) is the supplied voltage in \(V\).
• \(R\) is the electrical resistance in \({\Omega}\).
• \(D\) is the wire diameter, taken from manufacturing specifications in \(m\).
• \(\rho\) is the electrical resistivity of the wire material in \({\Omega m}\).
• \(l\) is the wire length in \(m\).

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Use this constraint when the heatup time of a region should be limited. The heatup time is the time it takes for the region to heat up from an initial temperature to a target temperature. Note that the target temperature does not have to be the final temperature the region reaches, it can be smaller.

Mathematical formulation

\[C= 5 \tau_c = \frac{5}{Q/(\rho c_p V (T_{initial} - T_{target}))} \leq t_{heatup}\]

where

• \(C\) is the value of the contraint in \(s\).
• \(t_{heatup}\) is the maximal heatup time in \(s\).
• \(\tau_c\) is the time constant in \(s\).
• \(Q\) is the heat supplied to the region in \({W}\).
• \(\rho\) is the density of the region material in \(kg/m^{3}\).
• \(c_p\) is the heat capacity of the reigon material in \(J/(kgK)\).
• \(V\) is the volume of the region in \(m^3\).
• \(T_{initial}\) is the initial temperature of the region in \(K\).
• \(T_{target}\) is the target temperature of the region in \(K\).