Patent Description:
Topology optimization (topology optimization) is a research field concerned with exploring a design space and deciding where to put material and, equally important, where not to. This allows designing complex, organic structures that would not result from a traditional optimization approach that aims to find the best parameters for a given model, since parametrization inherently limits the design space. Due to its complex designs, topology optimization forms a complementary technology pair with additive manufacturing (AM), which is capable of printing materials everywhere inside a 3D design space and thus enables the manufacturing of the complex/organic structures found by the topology optimization optimizer.

In other words: topology optimization is a method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizing the performance of the system.

For this invention, optimal may be regarded in the mechanical sense, i.e., structures with a layout/design that results in the lowest compliance or mass under prescribed loads and boundary conditions. Due to the flexibility of the proposed framework, expansion to other fields of physics is possible. An improvement is an amendment from a current configuration towards an optimization target respectively an optimum as explained.

Multiscale topology optimization is concerned with finding optimal structures on multiple, but usually two, scales. This means in practice that two optimization loops are used. The outer loop is the standard monoscale topology optimization framework that decides the optimal amount of material at every point in the design. The inner loop looks at a single point and, through an assumption of scale separation, uses homogenization to decide the optimal microstructure at this point.

Topology optimization according to the invention is a multiscale optimization method for finding.

Methods of topology optimization may be judged based on the three main challenges: efficiency, optimality, and manufacturability.

Efficiency is the easiest to judge but also the hardest to solve, since technically two-scale topology optimization requires a double optimization loop. Some methods severely reduce the microstructure design space to combat this problem or instead solve it in an offline phase.

Solving for each microstructure a full monoscale topology optimization in every iteration requires enormous computational effort. Literature after this has compromised optimality in favor of efficiency, mostly through parameterization of the microstructure and offline computations. Multiscale topology optimization methods can thus be judged on their optimality by looking at how severely the parametrization of their microstructures limits the design space. In this sense, monoscale TO methods are simply two-scale methods with a microstructure design space restricted to the density. In general, the less parametrization the more optimal the design will be (but at a higher cost).

Optimality respectively best improvement is one main advantage of this invention because the invention ensures optimality without compromising efficiency.

<CIT> deals with a method for improving the topology of a component as defined in the preamble of claim <NUM>.

<CIT> relates to the field of topology optimization.

An overview of multiscale topology optimization algorithms is given by Wu et al.

proposes to divide algorithms into 5x3=<NUM> categories based on the restrictions that are placed on.

They are thus divided into how their microstructures are parameterized, and this parametrization has major effects on the three challenges/criterions explained above.

For the restriction on density, Wu et al. make a division between.

As explained above, any restriction on density is suboptimal and it is one object of this invention to avoid this. Preferably category <NUM>) is therefore a decent option for multiscale topology optimization.

With respect to the restrictions on the set of admissible stiffness tensor, Wu et al. divide all multiscale optimization methods into five types.

The first uses optimal sets of stiffness tensors, which correspond to the theoretically known rank-N laminates (The rank-n laminate consists of alternating layers of the phase "-" and rank-(n - <NUM>) layers. Although, these microstructures are ensured to be optimal, they are not manufacturable since they exist on more than one scale.

The second type uses inverse homogenization: an optimization technique that tries to find microstructures with homogenized material properties that lie as close as possible to given, desired properties. It thus allows generating optimal microstructures without restrictions on their orientation, connectivity,. These methods thus provide very optimal designs but suffer from computational inefficiency.

The third type also uses inverse homogenization but puts some restrictions on the unit cell design. It's thus less optimal than the second type but usually more manufacturable.

The fourth and fifth type use unit cells parameterized with multiple parameters or simply the density, respectively. An example of the fourth type microstructures restricted to lattices where the beam diameters are used as parameters. In case all beam diameters are constrained to be equal, only one parameter (the beam diameter/relative density) remains and the method then belongs to the fifth type. Due to the aggressive parametrization, the resulting designs are ensured to be suboptimal.

Although not covered by Wu et al. , an important variation on the second type exists, namely the database- or Vademecum-enhanced methods from Ferrer et al. ) and Djourachkovitch et al.

Basically, instead of doing the microscale optimization loop online, causing computational inefficiency, a lot of microstructures are optimized offline. These optimized solutions are stored in a database (referred to as a Vademecum by Ferrer et al. ) and used during the macroscale optimization procedure. As such, the large online cost is replaced by an offline cost. Furthermore, costly re-computation of previously optimized structures is prevented. These solutions could thus be regarded as the state of the art, but currently they lack the final insight that unlocks their true capabilities. This final insight, namely that all optimal microstructures lie on a Pareto front, is exactly what this invention discloses and explained in the next section. The remaining part of this section discusses first the works of Ferrer et al. and second the paper of Djourachkovitch et al. and focusses on how their formulations limit the design space.

First, Ferrer et al propose a 'Vademecum-based' approach to multiscale topology optimization. A Vademecum of optimal microstructures is computed and used in the macroscale optimization loop. A large restriction is immediately placed on the optimal microstructures: they all have the same relative density (namely <NUM>%). Furthermore, since a Vademecum is basically a point cloud without structure, the macroscale algorithm simply loops through all entries and selects the best. This is moderately efficient and results in only a zeroth order approximation of the Pareto front that the Vademecum point cloud represents. With respect to manufacturability, Ferrer et al. propose a way to add manufacturing constraints by severely restricting their design space. In conclusion, severe restrictions are placed on optimality and manufacturability in order to ensure computational efficiency.

Second, Djourachkovitch et al. also propose a database-assisted strategy, although this time another inverse homogenization formulation is used. Although the microstructure's relative density is allowed to change, the microstructure layout is kept fixed everywhere in design space. This restriction is at least equally restrictive, if not more, than fixing the microstructure density but allowing it to change over the design domain. The main advantage of this technique is the manufacturability, which is severely improved when only a single microstructure exists. Since a database is used, again optimal structures are interpreted as a point cloud and this has severe complications for the method: a zeroth order algorithm is again used (since a database is non-smooth) and optimality is lost since the database, although large, is limited in size.

It is one object of the invention to overcome or at least mitigate the problems identified with the prior art as explained above.

To obtain a component design with optimal topology most efficiently and proximate to optimality the invention proposes a method of the incipiently mentioned type with the additional steps:.

The invention solves the material design problem by using a surrogate model relating cell stiffness tensors to a unit cell average density. This surrogate model preferably applies parameters corresponding to the degree of freedom of the underlying physical problem respectively the cell stiffness tensor. For homogenization to e.g., anisotropic materials, at most <NUM> parameters are required to parametrize the cell stiffness tensors.

One preferred embodiment provides that the elements of the stiffness tensor preferably all of them are improved or optimized. This results in a multiple objective improvement, and thus the formulation is a multi-objective optimization problem. This formulation to improve the stiffness tensor E may be written in a more simplified way as: <MAT>.

This formulation may be applied for optimizing the complete stiffness tensor respectively all its components at once (the constraints may be unchanged). This optimization problem has more than one objective and may consequently result in several solutions. Each of these solutions lies on a Pareto frontier. Specifying the importance of the objectives reduces the solution space significantly and may even lead to a single solution if the importance of each of your objectives is specified. One possibility to set a suitable optimization target is obtained by assigning an importance weight as a factor to each one of the tensor elements. Preferably this assignment may be combined with summing up all the weighted elements in order to maximize this sum. This beneficial approach scalarizes the optimization target changing the problem from multiple objectives to a single objective. This scaling and adding may be formulated as: <MAT>.

This process successfully improves the stiffness tensor elements in case that these are strictly convex. In case the stiffness tensor optimizing problem doesn't behave convexly the optimization may be performed by a suitable algorithm for multiobjective optimization, e.g., known from Mueller and Gritschneder (<NPL>. <NUM>/<NUM>).

According to a preferred embodiment an optimization may apply an algorithm which finds points of the Pareto front. Specifically, a suitable algorithms may explore a map from the importance weight vector w = [w<NUM>,<NUM>, w<NUM>,<NUM>,. , w<NUM>,<NUM>] (which specifies how important each objective is) to the corresponding performance vector f (in this case the stiffness tensor f = [E<NUM>,<NUM>, E<NUM>,<NUM>,. , E<NUM>,<NUM>].

The parametrization of the stiffness tensor using said surrogate model by said importance weights wi,j may be done for only every independent element <MAT> in the homogenized stiffness tensor. The parametrization may have the advantage of improved stability over specifying a set of desired material parameters, which may be unattainable. For homogenization to anisotropic materials, at most <NUM> weights are required to parametrize the homogenized stiffness tensor. These weights including a cell average density may be called hyperparameters of said surrogate model. Preferably said importance weights wi,j lie between <NUM> and <NUM> and they are required to sum to <NUM>: <MAT>.

The following formulation may then be obtained. <MAT> wherein this formulation is subject to constraints:.

Here, the optimization variable ρ is the standard microscopic density distribution inside a unit cell. This density distribution is used to maximize the objective function, a weighted sum of the elements of the homogenized stiffness tensor EH. This homogenized stiffness tensor EH is found by solving a standard homogenization procedure, which consists of solving the six boundary value problems corresponding to constraint <NUM>) (solid state mechanics; mechanical force equilibrium). Constraint <NUM>) and <NUM>) are volume and density constraints which may also be applied in monoscale topology optimization. Additional connectivity and manufacturability constraints may be added. Connectivity and manufacturability may be guaranteed by constraining the unit cell layout design library Vademecum to layouts which fulfill these requirements.

A preferred embodiment of the invention provides that providing a starting component design specifies at least one of the following parameters:.

Another preferred embodiment of the invention provides segmenting said component design into unit cells with the further steps of:.

Some of these quantities relating to solid state mechanics may be simply derived from the macroscale mechanical equilibrium (unit cell stiffness tensor, load case) and other may be set to an average value. FEM may be used to determine some of these quantities in more complex cases.

According to the invention the invention further provides generating said surrogate model applying the steps of:.

According to the invention said weighting-parameters are provided for every independent element of said unit cell stiffness tensor, such that for a given unit cell average density variations of said unit cell stiffness tensor are parameterized by said weighting-parameters.

Another preferred embodiment of the invention provides improving or optimizing at least one specific unit cell parameter using said surrogate model. This process may further comprise:.

Such improving may be done preferably for all unit cells. For each unit cell an optimal weighting parameter set may be determined resulting in a stiffness tensor which may be converted to a standard layout by inverse homogenization.

Another preferred embodiment of the invention provides changing said component design by amending said material mass distribution according to optimizing results such that:.

The re-distribution of the mass all over the component unit cells depending on their respective performance regarding the stiffness tensor efficiently leads the design process towards an optimum of mass distribution and microscale layout.

A preferred embodiment of the invention provides that said standard layouts for the unit cells are characterized by:.

This preselection of optimal layouts increases the efficiency of the process significantly.

According to the invention the invention provides that said surrogate model is configured such that it models said unit cell stiffness tensor for respectively predefined unit cell average densities as being located on a Pareto-front when keeping the sum of said weighting-parameters constant. The proposed technique solves the multiscale topology optimization problem using a surrogate model of microstructures lying on a Pareto front.

This idea of applying Pareto optimality overcomes issues of conventional multiscale topology optimization technique in particular regarding the challenge of optimality. Parametrizing the microstructure in any way normally sacrifices optimality due to not providing the full design space of possible microstructures. The invention solves this problem by a logical restriction - requiring that the employed microstructures are optimal "in some sense". When the optimizing target is mass reduction or least density the principle may be named as "do less with more". The proposed method follows such rule as: if a microstructure A can be replaced by another microstructure B that uses less material, then microstructure A is clearly sub-optimal. By repeating this process, only "optimal" microstructures are retained.

The theory surrounding Pareto efficiency is known from economic theory and philosophy. The act of replacing microstructure A by B is called a "Pareto improvement" and by continuously doing so (through optimization routines) a "Pareto Optimal", here a microstructure is obtained. The set of all Pareto optimal solutions is referred to as the "Pareto front (ier)".

The restriction that microstructures lie on a Pareto front is one essential understanding enabling efficiency. All conventional multiscale frameworks that employ inverse homogenization (a. a material design) generate microstructures. So far not realized is that these microstructures may be located on a Pareto front. Applying this understanding now enables using said surrogate model for optimizing allowing flexibility, efficiency and manufacturability.

For every choice of the so-called "hyperparameters", i.e. the maximum relative density ρmax and the "importance weights" wi,j, another Pareto optimal material or structure or microstructure may be obtained. By varying the hyperparameters, one can thus 'trace' the Pareto front, effectively obtaining a map to all "useful" microstructures. To decide which Pareto optimal microstructure is employed at every region k on the macroscale, the invention optimizes the hyperparameters <MAT> and <MAT> for every such region. Mathematically, given N macroscale regions, the following (discrete form of the) macroscale optimization problem is then solved. Basically, this process does not provide that the material-specification or exact layout (i.e. what your metamaterial looks like) is the output of the surrogate model optimizing process. Do note that the exact layout (i.e. what your metamaterial looks like) is obtained after the improving or optimizing step using inverse homogenization. The surrogate merely model maps.

to the (compromise) stiffness tensor of the optimal metamaterial with that weight. In that sense, the surrogate model contains the information "what could be achieved" but not (yet) "how do we achieve that".

A preferred embodiment of the invention provides said optimization target is one of:.

Different design tasks or technical disciplines can be better achieved by means of a specific optimization objective than with others. , reducing the compliance of said unit cell under said unit cell load case may be done as: <MAT> wherein this formulation is subject to constraints:.

Here, the objective function is the compliance, which depends indirectly on the optimization variables <MAT> and <MAT> through the displacement u and the state equation (=> constraint <NUM>)). Constraint <NUM>) reflects the macroscale volume constrained by the maximum macroscale relative density Pmax. Finally, constraint <NUM>) contains the microscale hyperparameter constraints.

Due to the above formulations, two-scale optimality is guaranteed. Since microstructures are not parametrized or restricted in any way the complete design space can be employed. The understanding that the Pareto front is a smooth function of the hyperparameters, allows these hyperparameters to be used as macroscale optimization variables: something unseen in current literature that allows flexibility in optimization solvers, adding extra constraints, expansion to other physics.

Due to the concurrent solving of N inverse homogenization problems in each macroscale iteration computational resources are needed. A key element to ensured efficiency is a surrogate model Ψ according to the invention. This surrogate model may be e.g. a neural network, kriging interpolation scheme or any other suitable model type that may map the hyperparameters to their corresponding Pareto optimal stiffness tensor. It may be noted that the optimal microstructural density distribution ρ is not used for the surrogate model, since inverse homogenization suffers from a uniqueness problem: multiple microstructural densities can have the same homogenized properties. The surrogate model's output is basically a stiffness tensor of the optimal metamaterial. This means that, after the final iteration has occurred and the optimal hyperparameters are found, one single inverse homogenization problem must be solved for every region in order to extract the optimal CAD model.

This is one key feature to combine optimality and efficiency.

The surrogate may preferably be trained on sampled points of the pareto front and provides an accurate, efficient and dif-ferentiable map Ψ from the hyperparameters to the optimal stiffness tensor Eopt. This means it also may provide a complete representation of all the microstructural tradeoffs, i.e., it may support to answer questions like:.

This understanding may be useful for an engineer to find the best design.

Compared to the conventional zeroth order database selection strategies (e.g. Ferrer et al. and Djourachkovitch et al. ) this approach and it's variations provide a considerable improvement. This benefit is enabled by the realization that inversely homogenized points lie on a Pareto frontier.

Another preferred embodiment of the invention provides flexibility of the proposed methodology and allows ensuring manufacturability. By adding general manufacturing constraints containing hyperparameters such as a maximum length scale or connectivity parameters to the microscale formulation, manufacturing constraints may be ensured in a flexible way. The invention offers flexibility with respect to manufacturing. Another point of flexibility is the employed physics. Although here the linear elastic problem is solved, one can straightforwardly apply the invention to for example fluid flow problems by envisioning a class of Pareto optimal microstructures that optimally guide fluids given a certain pressure field.

The invention may preferably benefit from the understanding that inversely homogenized microstructures lie on a Pareto frontier, and that this frontier can be parametrized preferably using the inverse homogenization's hyperparameters. The inverse homogenization is thus changed slightly, using importance weights instead of desired material properties, leading to a unique new material design formulation. The realization of Pareto optimality, together with the hyperparameter parametrization, then lead to a unique macroscale formulation where the hyperparameters are optimized. This framework thus ensures complete macro- and micro-scale optimality, a very important criterion. Furthermore, the realization also enables an essential feature of the invention: a (smooth) surrogate model that allows efficient online evaluation of the material design formulation. The invention's flexibility also allows the easy adoption of extra manufacturability constraints and expansion to other physics.

Embodiments of the invention are now described, by way of example only, with reference to the accompanying drawings, of which:.

The illustration in the drawings is in schematic form.

<FIG> shows a simplified flow diagram illustrating a method according to the invention for improving the topology of a component CPT. A first step (a) provides a component CPT load case LDC including boundary conditions BCD for said component CPT. Such component may be understood as a real-world object as the cantilever CTV shown in <FIG>. As an input to the process a step (b) provides a starting component design CDG. This design may comprise unit cell parameters VXP as: a unit cell geometry VGM, a unit cell load case VLC including unit cell boundary conditions VBC, a unit cell average density VAD, a unit cell stiffness tensor VST.

The initial component design CDG is used for segmenting said component design CDG into unit cells VXL in step (c). Based on unit cell VXL standard layouts and their respective properties (homogenized unit cell stiffness tensor VST, unit cell average density VAD) in step (d) a surrogate model SGM that relates these quantities to each other is generated. Said standard layouts STL are characterized by:.

Said surrogate model SGM parameterizes for a given unit cell average density VAD variations of said unit cell stiffness tensor VST.

In a subsequent step (e) said surrogate model SGM is used for improving or optimizing at least one specific unit cell parameter VXP for each unit cell VXL towards an optimization target OTG. Said optimization target OTG may be one of:.

In a step (f) changing said component design CDG by amending (MMD => MMD') a material mass distribution MMD according to optimizing results of step (e). This may be done according to the following rules:.

In a loop (g) steps (e), (f) may be repeated until a predefined criterium PDC is met. The criterium PDC may e.g. be that the change rate (change per loop) decreases below a certain threshold (ΔMMD<ε).

According to the improved or optimized design CDG a topology optimized component CMT may be machined MCH, e.g. by additive manufacturing resp. by a 3D-printing apparatus.

<FIG> shows five neighboring standard layouts STL respectively microstructures. The standard layouts STL of this example are optimized without considering their connectivity. Connectivity constrains may be made to ensure manufacturability.

<FIG> shows: Illustration of the inverse homogenization IHO, solving a material design problem. On the left-hand side, a homogenized microstructure HNL with unknown layout is symbolized. This homogenized microstructure HNL has desired microstructure properties relating to average density/volume ratio and stiffness behavior. Via inverse homogenization IHO a specific layout is determined providing the desired properties.

<FIG> shows: Visualization of a Pareto front PRF for minimization of two objectives J1, J2. The Pareto front PRF is the location of Pareto-optimal solutions POS. Finding points on the front, called Pareto Front tracing, is numerically expensive due to the need for an iterative optimization routine. The Pareto front PRF divides areas of solutions which are unattainable UAT from sub-optimal solutions SOP. The figure illustrates two improvements:.

<FIG> illustates an overview of a possible framework according to the invention.

The invention solves the material design problem by generating and applying a surrogate model E=Ψ(wij, ρmax) which is mapping model hyperparameters wij and maximum density ρmax to optimize the stiffness tensor. Instead of specifying a set of desired material parameters, which may lead to instability when these are unattainable, said importance weight wi,j for every independent element <MAT> in the homogenized stiffness tensor is specified. For homogenization to anisotropic materials, at most <NUM> weights are required. They lie between <NUM> and <NUM> and they are required to sum to <NUM>: <MAT>.

The following formulation is then obtained. <MAT> Subject to.

Here, the optimization variable ρ is the standard microscopic density distribution inside a unit cell. This density distribution is used to maximize the objective function, a weighted sum of the elements of the homogenized stiffness tensor E. This tensor is found by solving a standard homogenization procedure, which consists of solving the six boundary value problems corresponding to constraint <NUM>). Constraint <NUM>) and <NUM>) are standard volume and density constraints similar to those found in monoscale TO.

Claim 1:
Method, to be performed on a computer, for improving the topology of a component (CPT), comprising:
(a) providing a component (CPT) load case (LDC) including boundary conditions (BCD) for said component (CPT),
(b) providing a starting component design (CDG),
(c) segmenting said component design (CDG) into unit cells (VXL),
characterized by the additional steps:
(d) generating a surrogate model (SGM) that relates these quantities to each other:
- possible unit cell stiffness tensors (VST),
- a unit cell average density (VAD),
wherein for a given unit cell average density (VAD) variations of said unit cell stiffness tensor (VST) are parameterized by said surrogate model (SGM),
wherein generating a surrogate model (SGM) comprises:
- providing at least two standard layouts (STL) for the unit cells (VXL),
- on basis of said standard layouts (STL) provided generating said surrogate model (SGM)
wherein said surrogate model (SGM) relates the following quantities to each other:
- said unit cell stiffness tensor (VST),
- said unit cell average density (VAD),
- weighting-parameters (WIJ), wherein said weighting-parameters (WIJ) are provided for every independent element of said unit cell stiffness tensor (VST), such that for a given unit cell average density (VAD) variations of said unit cell stiffness tensor (VST) are parameterized by said weighting-parameters (WIJ), wherein said surrogate model (SGM) is configured such that it models said unit cell stiffness tensor (VST) for respectively predefined unit cell average densities (VAD) as being located on a Pareto-front when keeping the sum of said weighting-parameters (WIJ) constant,
(e) using said surrogate model (SGM) for improving or optimizing at least one specific unit cell parameter (VXP) for each unit cell (VXL) towards an optimization target (OTG),
(f) changing said component design (CDG) by amending a material mass distribution (MMD) according to optimizing results of step (e).