DE102020203312A1 - Method and device for simulating the component behavior of a component with a heterogeneous microstructure - Google Patents

Abstract

The invention relates to a method for performing a mechanical simulation of a component with a heterogeneous microstructure, the following steps being carried out for each integration point (x) of the component: Provision (S1) of network parameters ((zNj = 1,2,..., 2N) A ¯, (θi = 0.1,…, Nk = 1.2,… 2i) A¯) of a deep material network (1) depending on a microstructure of the relevant integration point (x), where the network parameters ((zNj = 1, 2,…, 2N) A¯, (θi = 0.1,…, Nk = 1.2,… 2i) A¯) are determined with the help of a trained data-based network parameter model depending on microstructure configuration parameters (A (x)), whereby the microstructure configuration parameters (A (x)) specify the microstructure of the integration point (x) in question, - Use (S4) the Deep Material Network (1) to determine a stress and a consistent tangent for each integration point (x); - Perform (S5, S6) the mechanical simulation based on the stresses and consistent tangents to the integra tion points (x).

Description

Technical area

The invention relates to a method for predicting the mechanical behavior of components with heterogeneous microstructures based on direct numerical multi-scale simulations in the context of the finite element method (FEM).

Technical background

During the design of a component with a heterogeneous microstructure, the shape of the component is designed in such a way that it fulfills specified requirements for its function, e.g. B. has sufficient resistance to a thermomechanical load in a specific application over a desired service life.

To simulate the mechanical behavior of a component with a heterogeneous microstructure, FEM processes coupled with micromechanical simulations are generally used at the component level. The latter are usually based on the so-called Lippmann-Schwinger equation and are solved with the help of parallel Fast Fourier Transformation techniques (FFT), as for example in the publication Spahn et al., “A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms ”, 2014, Computer Methods in Applied Mechanics and Engineering 268. Commercial software packages are available for both macroscopic component simulation and micromechanical microstructure simulation, e.g. Ansys, Abaqus, LS-Dyna, GeoDict.

Fully coupled FEM-FFT simulations are associated with considerable numerical effort, which is why decoupling is pursued. In these approaches, numerous micromechanical simulations are used to generate data and store them in a database, which are then specifically evaluated with regard to the component simulation, e.g. Model order reduction method in combination with principal component analysis (PCA), e.g. Fritzen, F., & Böhlke, T. (2013) . Reduced basis homogenization of viscoelastic composites. Composites Science and Technology, 76 (4), 84-91, https://doi.org/10.1016/j.compscitech.2012.12.012. For microstructures with strongly non-linear behavior, these approaches are also associated with a very high numerical effort.

In this context, so-called deep learning approaches are pursued as an alternative to this, which represent the data described above with the help of neural networks. Neural networks are already used for simple microstructure behavior that does not depend on the load history, see e.g. Yvonnet, et al., "Computational homogenization of nonlinear elastic materials using neural networks," International Journal for Numerical Methods in Engineering 104 (12), 2015 .

Furthermore, from Liu, W. et al., "A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials", "Computer Methods in Applied Mechanics and Engineering 345, 2019 a new machine learning approach is known, which establishes the use of Deep Material Networks (DMN). This can also map history-dependent behavior via a combination of concepts of machine learning and traditional modeling via constitutive laws, but so far it can only be used sequentially for individual microstructures and not for the simulation of components with Suitable for heterogeneously distributed microstructures across the component.

So far there is no approach that enables the behavior of components with heterogeneous and history-dependent microstructure to be efficiently simulated.

Disclosure of the invention

According to the invention, a method for performing a mechanical simulation of a component with a heterogeneous microstructure according to claim 1, a method for training a network parameter model and a device according to the independent claims are provided.

Further refinements are given in the dependent claims.

• - Bereitstellen von Netzwerkparametern eines Deep Material Networks abhängig von einer Mikrostruktur an jedem Integrationspunkt, wobei die Netzwerkparameter mithilfe eines trainierten datenbasierten Netzwerkparametermodells abhängig von Mikrostrukturkonfigurationsparametern bestimmt werden, wobei die Mikrostrukturkonfigurationsparameter die Mikrostruktur am betreffenden Integrationspunkt angeben, wobei das Netzwerkparametermodell trainiert ist, um Mikrostrukturkonfigurationsparametern Netzwerkparameter des Deep Material Networks zuzuordnen;
• - Verwenden des Deep Material Networks, um eine Spannung und eine konsistente Tangente für jeden Integrationspunkt zu bestimmen, wobei insbesondere abhängig von den Spannungen und konsistenten Tangenten iterativ, insbesondere mithilfe einer FEM, an den Integrationspunkten ein globales, in der Regel nichtlineares Residuum für das Verschiebungsfeld berechnet wird;
• - Durchführen der mechanischen Simulation basierend auf den Spannungen und konsistenten Tangenten an den Integrationspunkten.

According to a first aspect, a method for performing a mechanical simulation of a component with a heterogeneous microstructure is provided, the following steps being carried out:

• - Provision of network parameters of a deep material network depending on a microstructure at each integration point, the network parameters being determined with the aid of a trained data-based network parameter model depending on microstructure configuration parameters, the microstructure configuration parameters specifying the microstructure at the integration point in question, the network parameter model being trained to use microstructure configuration parameters of the Assign deep material networks;
• - Using the Deep Material Network to determine a stress and a consistent tangent for each integration point, depending in particular on the Stresses and consistent tangents iteratively, in particular with the help of an FEM, a global, usually non-linear residual for the displacement field is calculated at the integration points;
• - Perform the mechanical simulation based on the stresses and consistent tangents at the integration points.

Initialization of the state variables (i.e. the internal variables for describing the non-linear behavior of the virtual phases such as plastic strains) in the DMN, which describe the history-dependent behavior. To calculate the displacement field in the component with the help of the FEM, the material law, which determines the local stress of the material from the local strain (and its history), is determined with the help of the DMN.

⊂ ℝ³ definiert. Mithilfe von herkömmlichen makroskopischen mechanischen Bauteilsimulationen wird die Impulsbilanz als partielle Differentialgleichung im Kontext der FEM gelöst, die Randbedingungen und räumliche oder zeitliche Anfangszustände berücksichtigt. Die Differentialgleichung hat dabei die starke Form

die für jeden Ort x und jeden Zeitpunkt t abhängig von der externen Belastung y(x,t) eine Gleichgewichtsbedingung darstellt. Die Spannungen σ(x,t) beschreiben dabei die individuelle Materialantwort.Basically, a macroscopic component is in a three-dimensional space as the set of points x ∈

⊂ ℝ ³ defined. With the help of conventional macroscopic mechanical component simulations, the momentum balance is solved as a partial differential equation in the context of the FEM, the boundary conditions and spatial or temporal initial states are taken into account. The differential equation has the strong form

the one for every place x and each point in time t depending on the external load y ( x , t) represents an equilibrium condition. The tension σ ( x , t) describe the individual material response.

die den symmetrische Gradient des zunächst unbekannten Verschiebungsfeldes ε(x, t) = ∇_s u(x,t) und den lokalen Mikrostrukturkonfigurationsparameter A(x) auf die resultierenden Spannungen in ggf. nichtlinearer Weise abbildet. Im Kontext kurzglasfaserverstärkter Kunststoffe beschreibt der Mikrostrukturkonfigurationsparameter A(x) beispielsweise wie in einem mikroskopischen Kontrollvolumen die Fasern orientiert sind, in welchem Verhältnis das Faservolumen zum gesamten Kontrollvolumen steht und welche geometrische Form die Fasern aufweisen. Wenn für das Material eines Bauteils die konstitutive Antwort auf makroskopischer Ebene in dieser Form formuliert werden kann, kann das Verschiebungsfeld des Bauteils mit Hilfe der FEM durch die Lösung eines nichtlinearen Gleichungssystems bestimmt werden, z.B. mit Hilfe der Newton-Raphson Methode.A typical macroscopic simulation is based on the assumption of a functional dependence of stress on strain ε and microstructure configuration parameters A. in the shape $$\overline{\sigma }\left(\overline{x},t\right)=\sigma \left(\overline{\epsilon }\left(\overline{x},t\right);\overline{\mathrm{A.}}\left(\overline{x}\right)\right)$$

which is the symmetrical gradient of the initially unknown displacement field ε ( x , t) = ∇ _s u ( x , t) and the local microstructure configuration parameter A. ( x ) maps onto the resulting stresses in a possibly non-linear manner. In the context of short glass fiber reinforced plastics, this describes the microstructure configuration parameters A. ( x ) For example, how the fibers are oriented in a microscopic control volume, the relationship between the fiber volume and the total control volume and the geometric shape of the fibers. If the constitutive answer for the material of a component can be formulated in this form on a macroscopic level, the displacement field of the component can be determined with the help of the FEM by solving a nonlinear system of equations, e.g. with the help of the Newton-Raphson method.

Dieses Verfahren, das auch als FE² oder FEM-FFT bezeichnet wird, ist rechnerisch sehr aufwändig, da eine Vielzahl von Simulationen durchgeführt werden müssen (eine makroskopische und unzählige mikroskopische).If the formulation of a single-scale constitutive model is not possible, e.g. due to a complex microstructure, the stresses in the component must be calculated with explicit consideration of the behavior of this microstructure. The basic approach is at every point of the component x _I ∈ B. that must be evaluated for the FEM (ie at the integration points x _I ) to carry out another FEM or FFT simulation and use it to calculate the stresses and the consistent tangents. This is done at every point x _I ∈ B. a microstructure with the respective microstructure configuration parameter A. ( x _I ), which is referred to as the representative volume element (RVE). The macroscopically predominant elongations at the location of the RVE ε ( x _I , t) are now applied to the RVE with the help of suitable boundary conditions (usually periodic) and the stress distribution σ (x, t) for x∈RVE with the help of the constitutive models of the individual phases (i.e. the different materials that make up the microstructure exists, such as fiber and matrix in fiber-reinforced composite materials) of the RVE (and possibly their interaction, e.g. in the form of delamination) determined via an FEM or FFT calculation. The macroscopically effective tension σ ( x _I , t) can then be calculated as via an integration (homogenization) $$\overline{\sigma }\left(\overline{x},t\right)=\frac{1}{V_{\mathrm{R.}V\mathrm{E.}}}\underset{\mathrm{R.}V\mathrm{E.}\left({\overline{x}}_{\mathrm{I.}},\overline{\mathrm{A.}}\left({\overline{x}}_{\mathrm{I.}}\right)\right)}{\int }\sigma \left(\mathbf{x},t\right)\text{d}x$$

This method, which is also known as FE ² or FEM-FFT, is computationally very complex, since a large number of simulations have to be carried out (one macroscopic and countless microscopic).

Further developments are based on a number of micromechanical simulations that are decoupled from the component simulation to generate a lot of response data, which are then post-processed with the help of model order reduction methods such as PCA in order to derive effective models that are then used in a decoupled component simulation.

Another possibility is instead of formulating a physically based constitutive model σ ( x , t) = σ ( ε ( x , t); A. ( x )) to use a Deep Material Network (DMN), which can predict the local stress of the material with a complex microstructure and thus the microscopic FE or FFT simulations are superfluous power. The DMN is determined by its architecture (number of layers) as well as by the parameters (weights and rotations), which are identified with the help of elastic training data on an RVE. The DMN is therefore dependent on a constant microstructure configuration parameter A. .

• - Generierung synthetischer Mikrostrukturen unter Vorgabe von Mikrostrukturkonfigurationsparametern A(x): Mit Hilfe geeigneter Softwaretools (z.B. Schneider, M., „The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics.“, Computational Mechanics, 59, 247-263 2017 ) werden synthetisch dreidimensionale Bilddaten generiert, die ähnliche Eigenschaften wie die reale Mikrostruktur haben.
• - Generierung Trainingsdaten: Basierend auf der synthetischen Mikrostruktur werden zahlreiche elastische, zeitunabhängige Simulationen durchgeführt, wobei die Steifigkeiten der Phasen und damit auch die elastischen Kontraste (d.h. die Verhältnisse der elastischen Eigenschaften der Phasen) variiert werden. Input sind die (anisotropen) elastischen Steifigkeiten der Phasen und das Ergebnis ist die elastische, richtungsabhängige Steifigkeit des Verbundwerkstoffs an jedem Integrationspunkt.
• - Offline: Training des Deep Material Networks: Die numerisch erzeugten Trainingsdaten werden verwendet, um die DMN-Netzwerkparameter zu trainieren, die sich für ein DMN mit N Schichten aus ${{\displaystyle z}}_N^{j=\mathrm{1,2,}\dots {\mathrm{,2}}^N}$
  
  Gewichten und ${{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}$
  
  Rotationen zusammensetzen. Das trainierte DMN kann damit für beliebige Kombinationen von Steifigkeiten der einzelnen Phasen die richtungsabhängige effektive Steifigkeit der Mikrostruktur vorhersagen.
• - Online: Vorhersage zeitabhängiges Verhalten der Mikrostruktur: Das elastisch antrainierte DMN kann in einem weiteren Schritt nun erweitert werden um das nichtlineare, zeitabhängige Verhalten des Verbundwerkstoffes vorherzusagen. Hierzu werden die einzelnen Phasen in der untersten Schicht des DMNs als inelastische Phasen interpretiert und ihr (von der Belastungsgeschichte abhängiger) Zustand entsprechend mit internen Variablen versehen, die die zeitliche Entwicklung einer bestimmten Größe beschreiben (z.B. plastische Dehnungen, Schädigung, ...) und deren zeitliche Entwicklung durch eine gewöhnliche Differentialgleichung modelliert wird.

The structure and mode of operation of DMNs are described in Liu et al. [2018]. In general, a DMN describes the non-linear mechanical behavior of a particular microstructure, which is determined by the microstructure configuration parameters A. ( x ) is characterized for a volume element x̅. The application of the DMN to a specific microstructure can be divided into the following work steps:

• - Generation of synthetic microstructures by specifying microstructure configuration parameters A. ( x ): With the help of suitable software tools (e.g. Schneider, M., "The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics.", Computational Mechanics, 59, 247-263 2017 ) synthetic three-dimensional image data are generated which have properties similar to the real microstructure.
• - Generation of training data: Based on the synthetic microstructure, numerous elastic, time-independent simulations are carried out, whereby the stiffness of the phases and thus also the elastic contrasts (ie the proportions of the elastic properties of the phases) are varied. The input is the (anisotropic) elastic stiffness of the phases and the result is the elastic, direction-dependent stiffness of the composite material at each integration point.
• - Offline: Training of the Deep Material Network: The numerically generated training data is used to train the DMN network parameters that are specific to a DMN with N layers ${{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}$
  
  Weights and ${{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}$
  
  Assemble rotations. The trained DMN can thus predict the direction-dependent effective stiffness of the microstructure for any combination of stiffnesses of the individual phases.
• - Online: Prediction of time-dependent behavior of the microstructure: The elastically trained DMN can now be expanded in a further step to predict the non-linear, time-dependent behavior of the composite material. For this purpose, the individual phases in the lowest layer of the DMN are interpreted as inelastic phases and their state (depending on the load history) is provided with internal variables that describe the development of a certain variable over time (e.g. plastic strain, damage, ...) and whose development over time is modeled by an ordinary differential equation.

und ${{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}$

eines DMNs von der Topologie Mikrostruktur- gegeben durch den Parameter A(x) - ab, die sie beschreiben, so dass ${\left({{\displaystyle z}}_N^{j=\mathrm{1,2,}\dots {\mathrm{,2}}^N}\right)}_{\overline{A}}$

und ${\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{\overline{A}}.$

Damit DMNs in einer kompletten Bauteilsimulation für die Vorhersage des Mikrostrukturverhaltens an allen Stellen des Bauteils verwendet werden können, müssen entsprechend für alle im Bauteil auftretenden Mikrostrukturen Trainingsdaten generiert und das DMN antrainiert werden. Die makroskopische Bauteilsimulation erfolgt dann entsprechend folgendem Ablauf:

• - Zunächst werden die internen Variablen der inelastischen Phasen (das sind die internen Variablen zur Beschreibung des nichtlinearen Verhaltens der virtuellen Phasen wie z.B. plastische Dehnungen) aller DMNs initialisiert.
• - Wird nun eine makroskopische Belastung/Randbedingung auf dem Bauteil inkrementell (inkrementell, da bei nichtlinearen Problemen die Lösung vom Belastungspfad abhängig ist und stückweise - also inkrementell - aufgebracht werden muss) aufgebracht, wird in jedem Zeitschritt im Sinne des iterativen Verfahrens und/oder zur Berücksichtigung der Belastungshistorie folgende Schritte ausgeführt:
  • ◯ Im Kontext der FEM wird in jedem Zeitschritt das Verschiebungsfeld gelöst, das die obige Gleichgewichtsbedingung erfüllt.
  • ◯ Um dies zu erreichen, muss das DMN, unter Berücksichtigung der lokalen Mikrostruktur, ausgewertet werden um die Spannungen und die konsistente Tangente zu liefern. Hierzu wird die makroskopische Dehnung ε(x _I, t) entsprechend der trainierten DMN-Parameter auf die einzelnen Phasen in der untersten Schicht des DMNs verteilt.
  • ◯ Je nach Beanspruchung (Belastung) der einzelnen Phasen wird ein Update der internen Variablen durchgeführt. Danach können dann die Spannungen und die konsistenten Tangenten der einzelnen Phasen berechnet werden.
  • ◯ Die gesamte makroskopische Spannung und konsistente Tangente wird über die Phasen entsprechend der Gewichte aufsummiert.

As described above, the parameters depend ${{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}$

and ${{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}$

of a DMN from the topology microstructure - given by the parameter A. ( x ) - from which they describe so that ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}}$

and ${\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}.$

So that DMNs can be used in a complete component simulation for predicting the microstructure behavior at all points of the component, training data must be generated for all microstructures occurring in the component and the DMN trained. The macroscopic component simulation then takes place according to the following sequence:

• - First, the internal variables of the inelastic phases (these are the internal variables for describing the non-linear behavior of the virtual phases such as plastic strains) of all DMNs are initialized.
• - If a macroscopic load / boundary condition is now applied to the component incrementally (incrementally, since the solution is dependent on the load path in the case of non-linear problems and has to be applied piece by piece - i.e. incrementally), in every time step in the sense of the iterative method and / or for Taking into account the load history, the following steps are carried out:
  • ◯ In the context of the FEM, the displacement field that fulfills the above equilibrium condition is solved in each time step.
  • ◯ To achieve this, the DMN must be evaluated, taking into account the local microstructure, in order to provide the stresses and the consistent tangent. For this purpose, the macroscopic elongation ε ( x _I , t) distributed to the individual phases in the lowest layer of the DMN according to the trained DMN parameters.
  • ◯ An update of the internal variables is carried out depending on the use (load) of the individual phases. Then the stresses and the consistent tangents of the individual phases can be calculated.
  • ◯ The total macroscopic stress and consistent tangent is added up over the phases according to the weights.

Because different microstructure configuration parameters A. lead to different DMNs (DMN (.4)) and there the microstructure and thus A. can vary greatly within a component (e.g. injection molding of short glass fiber reinforced plastics), a very large number of DMNs may have to be trained for this approach, which in turn requires a large number of RVEs and elastic training calculations.

und ${\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{\overline{A}}$

für alle Mikrostrukturen zu erzeugen), sondern einen Trainingsdatensatz DMN(A) für eine Trainingsmenge A zu erzeugen und damit ein datenbasiertes Netzwerkparametermodell zu trainieren, das den Zusammenhang von A mit den DMN Parametern vorhersagen kann und zur Parametrierung von weiteren DMNs verwendet werden kann, deren Mikrostrukturkonfigurationsparameter A außerhalb des Trainingsdatensatzes liegen.This invention now consists in not having the DMNs different for each A. to retrain (and thus the corresponding network parameters ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}}$

and ${\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}$

for all microstructures), but a training data set DMN ( A. ) for a training set A. to generate and thus to train a data-based network parameter model that shows the relationship between A. can predict with the DMN parameters and can be used to parameterize other DMNs, their microstructure configuration parameters A. lie outside the training data set.

• ◯ Zunächst wird der betrachtete Mikrostrukturparameterraum (also z.B. der Raum aller Mikrostrukturparameter, die faserverstärkte Kunststoffe beschreiben) mit einer finiten Anzahl A _α diskretisiert, die den Raum ausreichend abdecken.
• ◯ Für jeden dieser Mikrostrukturparameter A _α wird nun eine repräsentative Mikrostruktur erzeugt und mit Hilfe von FFT-Rechnungen mit verschiedenen Steifigkeitskontrasten je ein DMN trainiert, für das die Netzwerkparameter dann ${\left({{\displaystyle z}}_N^{j=\mathrm{1,2,}\dots {\mathrm{,2}}^N}\right)}_{{\overline{A}}_{\alpha }}$
  
  und ${\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{{\overline{A}}_{\alpha }}$
  
  sind.
• ◯ Die Ergebnisse der Berechnungen werden nun zusammengefasst als Datensatz $\left\{{\overline{A}}_{\alpha }\to {\left({{\displaystyle z}}_N^{i=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{{\overline{A}}_{\alpha }},{\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{{\overline{A}}_{\alpha }}\right\}$
  
  der jedem Mikrostrukturparameter A _α einen DMN-Parametersatz ${\left({{\displaystyle z}}_N^{j=\mathrm{1,2,}\dots {\mathrm{,2}}^N}\right)}_{{\overline{A}}_{\alpha }},{\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{{\overline{A}}_{\alpha }}$
  
  zuordnet.
• ◯ Dieser Datensatz wird verwendet, um einen geeigneten Regressionsalgorithmus zu trainieren - z.B. ein neuronales Netz.

The following steps are carried out for the training:

• ◯ First of all, the microstructure parameter space considered (e.g. the space of all microstructure parameters that describe fiber-reinforced plastics) is given a finite number A. _α discretized that sufficiently cover the space.
• ◯ For each of these microstructure parameters A. _α a representative microstructure is now generated and, with the help of FFT calculations with different stiffness contrasts, a DMN is trained for each of the network parameters ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}$
  
  and ${\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}$
  
  are.
• ◯ The results of the calculations are now summarized as a data set $\left\{{\overline{\mathrm{A.}}}_{\alpha }\to {\left({{\displaystyle z}}_N^{i=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}\right\}$
  
  of each microstructure parameter A. _α a DMN parameter set ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}$
  
  assigns.
• ◯ This data set is used to train a suitable regression algorithm - e.g. a neural network.

Einmal trainiert, können damit hocheffizient beliebig viele DMNs parametriert werden, die für Bauteilsimulationen mit beliebigen Verteilungen an Mikrostrukturen aus der zugrundeliegenden Klasse verwendet werden können.The trained regression algorithm now also enables new A. ≠ A. _α a prediction of associated DMN parameters ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}.$

Once trained, any number of DMNs can be parameterized in a highly efficient manner, which can be used for component simulations with any distribution of microstructures from the underlying class.

After generating the DMNs of the underlying microstructures using the regression algorithm, the procedure is the same as for classic DMNs.

Since the DMNs only describe the topology of the microstructures and the non-linear behavior of the phases is modeled independently of this, the regression algorithm must be used to predict the DMN parameters A. can only be trained once for a microstructure class and can then be used independently of variations in the material models of the individual phases.

Figure list

• 1 eine schematische Darstellung eines Deep Material Networks mit den Netzwerkparametern, die durch Aktivierungen z_j und Rotationswinkel θ_j angegeben sind;
• 2 ein Flussdiagramm zum Durchführen einer mikromechanischen Simulation eines Bauteils basierend auf dem Deep Material Networks der 1; und
• 3 ein Verfahren zum Trainieren eines datenbasierten Netzwerkparametermodells zum Zuordnen von Netzwerkparametern des Deep Material Networks zu einer bestimmten charakterisierten Mikrostruktur.

Embodiments are explained in more detail below with reference to the accompanying drawings. Show it:

• 1 a schematic representation of a deep material network with the network parameters indicated _{by activations z j} and rotation angle θ _j;
• 2 a flowchart for performing a micromechanical simulation of a component based on the deep material network of FIG 1 ; and
• 3 a method for training a data-based network parameter model for assigning network parameters of the deep material network to a specific characterized microstructure.

Description of embodiments

∈ ℝ³. Mithilfe einer partiellen Differentialgleichung und vorgegebenen räumlichen und zeitlichen Anfangszuständen kann das mechanische Gleichgewicht des Bauteils abhängig von der externen Belastung γ(x, t) auf das Bauteil, z.B. angegeben als Krafteinwirkung, eine lokale Mikrostrukturkonfiguration, die durch Mikrostrukturkonfigurationsparameter A(x) für das lokale Volumenelement des betreffenden Integrationspunkts x̅ und für einen Zeitpunkt t angegeben ist, und die Spannungen σ(x, t) für das lokale Volumenelement des betreffenden Integrationspunkts x und für einen Zeitpunkt t, die die individuelle Materialantwort darstellt, ausgedrückt werden:

Typische einskalige Simulationen gehen von einer funktionalen Abhängigkeit in der Form $$\overline{\sigma }\left(\overline{x},t\right)=\overline{\sigma }\left(\overline{\epsilon }\left(\overline{x},t\right);\overline{A}\left(\overline{x}\right)\right)$$

bezogen auf die Dehnung ε(x, t), die dem symmetrischen Gradienten ε(x, t) = ∇_s u(x,t) entspricht, und der lokalen Mikrostrukturkonfiguration auf die resultierende Spannung gemäß einem nichtlinearen Verhalten aus. Setzt man das Materialgesetz in die Differentialgleichung ein, kann man die Gleichung (numerisch) nach den Verschiebungen u(x, t) lösen und erhält daraus die Deformation und Spannungsverteilung im Bauteil.In principle, the numerical simulation of the mechanical behavior of a component takes place within the framework of the FEM by subdividing the component into volume elements, the mechanical properties of which are based on an integration point assigned to the volume element x̅ be calculated. The integration points x̅ are located in a three-dimensional space according to the component geometry

∈ ℝ ³ . With the help of a partial differential equation and given spatial and temporal initial states, the mechanical equilibrium of the component can be dependent on the external load γ ( x , t) a local microstructure configuration on the component, e.g. specified as the force effect, which is determined by microstructure configuration parameters A. ( x ) for the local volume element des relevant integration point x̅ and given for a point in time t, and the stresses σ ( x , t) for the local volume element of the integration point in question x and for a point in time t, which represents the individual material response, the following can be expressed:

Typical single-scale simulations assume a functional dependency in the form $$\overline{\sigma }\left(\overline{x},t\right)=\overline{\sigma }\left(\overline{\epsilon }\left(\overline{x},t\right);\overline{\mathrm{A.}}\left(\overline{x}\right)\right)$$

related to the elongation ε ( x , t) corresponding to the symmetrical gradient ε (x, t) = ∇ _s u ( x , t), and the local microstructure configuration on the resulting stress according to a non-linear behavior. If you insert the law of material into the differential equation, you can use the equation (numerically) after the displacements u ( x , t) and derives from this the deformation and stress distribution in the component.

To do the calculation based on the microstructure configuration parameters A. ( x ), a combined system of deep material network and data-based network parameter model is used during the macroscopic calculation phase, which includes the stress response of the material taking into account complex heterogeneous microstructures.

For this calculation, the sensitivity of the stresses in relation to the strain increments is given by D. ( ε ( x , t); A. ( x )), Part of the inference phase and should be provided by the trained network parameter model. The sensitivity of the stress in relation to the strain increment corresponds to the total stiffness D of the volume element in question, which in the case of heterogeneous materials can be determined in a known manner from the individual stiffnesses of the materials used in the heterogeneous material with the help of the configured deep material network.

In 1 For example, a deep material network 1 with the weights (activations) z _j and the rotation angles θ _{j is} specified for two material components, for example, in order to use corresponding stiffness tensors D ₁ and D _{2 to obtain an overall stiffness tensor D rve} valid for the volume element in question for the further calculation according to to provide the above simulation rule. The overall stiffness for the volume element depends on the microstructure configuration parameters A. ( x ) of the relevant volume element (indicated by the integration point x̅), which are specified for a component to be simulated.

2 shows a flow chart to illustrate a method for simulating a mechanical property of a component. The simulation takes place on a conventional data processing device, whereby the following algorithm can be implemented as hardware or software.

For the simulation, step S1 the elastic stiffnesses D of the various material components as well as the initial values for local internal states of the virtual phase are given. The virtual phases correspond to the composition of the material of the component in a relevant volume element and parameters which can characterize the distribution of the materials in the volume element.

Furthermore, in step S2 Microstructure configuration parameters A. ( x ), which characterize the microstructural structure of the volume element.

eines Deep Material Networks 1 ermittelt. Das Netzwerkparametermodell ist trainiert, um einer durch die Mikrostruktur-Konfigurationsparameter A(x) charakterisierten Mikrostruktur Netzwerkparameter $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1,2,}\dots {\mathrm{,2}}^N}\right)}_{\overline{A}},{\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{\overline{A}}\right)$

des DMN-Netzwerks zuzuordnen.In step S3 are network parameters using a trained network parameter model $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right)$

of a deep material network 1 determined. The network parameter model is trained to be one through the microstructure configuration parameters A. ( x ) characterized microstructure network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right)$

of the DMN network.

In step S4 the Deep Material Network configured in this way is used to generate a given macroscopic strain increment Δ ε ( x ) carry out a backpropagation step to determine the expansion increments Δ ε ( x ) of the individual virtual phases in the lowest layer of the Deep Material Network 1 using a Newton-Raphson iteration.

Then in step S5 these expansion increments Δ ε ( x ) is used to update the internal states for the relevant integration point (e.g. the plasticity) (by adding up the expansion increments Δ ε ( x ) to the strain present for the respective volume element, the respective phase ε ( x ) and compute the tangent based on the elastic stiffness and inelastic updates. The tangent corresponds to the derivation of the stresses after the elongations. There is a linear, elastic component that corresponds to the elastic stiffness. At the same time, however, there is also a non-linear component: If the change plastic strains, this also has an influence on the tension. The tangent combines both effects.

In step S6 forward propagation through the DMN is now dependent on the expansion present for the respective volume element and the phase in question ε ( x ) performed to a total stiffness D. ^rve ( x ) for the volume element.

This procedure is carried out for each of the integration points x̅.

In order to determine the network parameters for the DMN, a data-based network parameter model is provided which, with the aid of one of the flowcharts in FIG 3 illustrated method can be trained.

${{\displaystyle \overline{D}}}_2^i,$

... mit i=1,...,M und repräsentativen Mikrostrukturen mit den Mikrostrukturkonfigurationseigenschaften A_a mit a=1,...,L bereitgestellt.In step S11 a number M of stiffness tensor combinations are used for this ${{\displaystyle \overline{\mathrm{D.}}}}_1^i,$

${{\displaystyle \overline{\mathrm{D.}}}}_2^i,$

... with i = 1, ..., M and representative microstructures with the microstructure _{configuration properties A a} with a = 1, ..., L provided.

der Mikrostrukturen mit den jeweiligen Mikrostrukturkonfigurationsparameter A _α und die einzelnen Phasen mit den Steifigkeiten ${{\displaystyle \overline{D}}}_1^i,{{\displaystyle \overline{D}}}_2^i,\dots \text{z}\text{.B}\text{.}$

... mithilfe einer Fast-Fourier-Transformation in an sich bekannter Weise ermittelt.Then in step S12 the resulting effective elastic stiffness tensors ${{\displaystyle \overline{\mathrm{D.}}}}_{\alpha }^i,$

of the microstructures with the respective microstructure configuration parameters A. _α and the individual phases with the stiffnesses ${{\displaystyle \overline{\mathrm{D.}}}}_1^i,{{\displaystyle \overline{\mathrm{D.}}}}_2^i,...\text{z}\text{.B}\text{.}$

... determined with the help of a Fast Fourier Transformation in a manner known per se.

bei denen die Steifigkeitstensorkombinationen ${{\displaystyle \overline{D}}}_1^i,{{\displaystyle \overline{D}}}_2^i,$

... auf den resultierenden effektiven elastischen Steifigkeitstensor ${{\displaystyle \overline{D}}}_{\alpha }^i,$

für die jeweilige durch A _α angegebene Mikrostruktur abgebildet werden.This results in training data sets ${\left\{{{\displaystyle \overline{\mathrm{D.}}}}_1^i,{{\displaystyle \overline{\mathrm{D.}}}}_2^i,\to {{\displaystyle \overline{\mathrm{D.}}}}_{\alpha }^i,\right\}}_{{\overline{\mathrm{A.}}}_{\alpha }},$

where the stiffness tensor combinations ${{\displaystyle \overline{\mathrm{D.}}}}_1^i,{{\displaystyle \overline{\mathrm{D.}}}}_2^i,$

... on the resulting effective elastic stiffness tensor ${{\displaystyle \overline{\mathrm{D.}}}}_{\alpha }^i,$

for the respective through A. _α specified microstructure can be mapped.

und ${\left({{\displaystyle \theta }}_{i=\mathrm{0,1,}\dots ,N}^{k=\mathrm{1,2,}\dots {\mathrm{,2}}^i}\right)}_{{\overline{A}}_{\alpha }}$

für die L verschiedenen Mikrostrukturen ermittelt, wobei N der Anzahl der Schichten des DMN entspricht. Hierzu eignet sich ein Backpropagation- oder ein Stochastic Gradient Descent Verfahren.This will be in step S13 now the network parameters of the Deep Material Network ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}$

and ${\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}$

determined for the L different microstructures, where N corresponds to the number of layers of the DMN. A back propagation or a stochastic gradient descent method is suitable for this.

wobei die Mikrostrukturkonfigurationsparameter A_a auf die Netzwerkparameter des Deep Material Networks mithilfe des Netzwerkparametermodells abgebildet werden.From this, one obtains for each of the microstructure configuration parameters specified by the microstructure A. _α specified microstructures have their own deep material network ${{\displaystyle \mathrm{D.}\mathrm{M.}N}}_{\left({{\displaystyle z}}_N^j,{{\displaystyle \theta }}_j^k\right)}^{{\overline{\mathrm{A.}}}_{\alpha }},$

wherein the microstructure configuration parameters A _a are mapped to the network parameters of the deep material network with the aid of the network parameter model.

der die Mikrostrukturkonfigurationsparameter A _α auf die Netzwerkparameter des DMN abbildet, wird in Schritt S14 das Netzwerkparametermodell in an sich bekannter Weise, z.B. mithilfe eines Backpropagation-Verfahrens, trainiert. Das Netzwerkparametermodell kann als Regressionsmodell, wie z.B. als Gaußprozessmodell oder als künstliches neuronales Netz ausgebildet sein.Using the training data set $\left\{{\overline{\mathrm{A.}}}_{\alpha }\to {{\displaystyle z}}_N^{i=\mathrm{1.2,}...{\mathrm{,\; 2}}^N},{{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right\},$

which is the microstructure configuration parameters A. _{α is mapped} to the network parameters of the DMN in step S14 the network parameter model is trained in a manner known per se, for example with the aid of a backpropagation method. The network parameter model can be designed as a regression model, such as a Gaussian process model or as an artificial neural network.

The network parameters of the deep material network can thus be predicted directly for each volume element by means of the microstructure _{configuration parameters A a, ie} ${\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{{\overline{\mathrm{A.}}}_{\alpha }}=NN\left({\overline{\mathrm{A.}}}_{\alpha }\right).$

The network parameter model trained in this way can then be used incrementally for the individual integration points for the micromechanical simulation of a component described above.

QUOTES INCLUDED IN THE DESCRIPTION

This list of the documents listed by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Non-patent literature cited

• Model order reduction method in combination with principal component analysis (PCA), e.g. Fritzen, F., & Böhlke, T. (2013) [0004]
• Yvonnet, et al., "Computational homogenization of nonlinear elastic materials using neural networks", International Journal for Numerical Methods in Engineering 104 (12), 2015 [0005]
• M., "The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics.", Computational Mechanics, 59, 247-263 2017 [0017]

Claims (8)

Method for performing a mechanical simulation of a component with a heterogeneous microstructure, whereby for each integration point ( x ) of the component, the following steps are carried out: - Provision (S1) of network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right)$

of a deep material network (1) depending on a microstructure of the relevant integration point ( x ), the network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right)$

using a trained data-based network parameter model depending on microstructure configuration parameters ( A. ( x )), with the microstructure configuration parameters ( A. ( x )) the microstructure of the integration point in question ( x ), - Using (S4) the Deep Material Network (1) to find a stress and a consistent tangent for each integration point ( x ) to determine; - Performing (S5, S6) the mechanical simulation based on the stresses and consistent tangents at the integration points ( x ). Procedure according to Claim 1 , whereby the mechanical simulation is carried out using a finite element method. Procedure according to Claim 2 , where for each of the integration points ( x ) from the finite element simulation a macroscopic strain increment (Δ ε ( x )) is determined and a backpropagation method is carried out in the DMN network to determine the expansion increments (Δ ε ( x )) of the virtual phase on the lowest level of the DMN network, using the expansion increments (Δ ε ( x )) of the virtual phase, the internal states that occur at the integration points ( x ) are available, with the help of the DMN the resulting homogenized total stiffness of the relevant integration point ( x ) is calculated by forward propagation. Method for training a network parameter model for use in a mechanical simulation of a component with a heterogeneous microstructure, with the following steps: - Provision of a number of microstructure configuration parameters ( A. ( x )) that define a space of microstructure parameters of possible heterogeneous microstructures; - Generation of a representative microstructure and training of a DMN with the help of FFT calculations with different stiffness contrasts for each microstructure configuration parameter ( A. ( x )) to the network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right)$

of the DMN; - Training the network parameter model $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right),$

around the microstructure configuration parameters ( A (x )) on the network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...{\mathrm{,\; 2}}^i}\right)}_{\overline{\mathrm{A.}}}\right)$

of the DMN. Method according to one of the Claims 1 until 4th , wherein the data-based network parameter model is designed as a regression model or a neural network. Device for performing a mechanical simulation of a component with a heterogeneous microstructure, the device being designed to provide for each integration point ( x ) of the component to perform the following steps: - Provision of network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...2^i}\right)}_{\overline{\mathrm{A.}}}\right)$

of a deep material network (1) depending on a microstructure of the integration point in question, the network parameters $\left({\left({{\displaystyle z}}_N^{j=\mathrm{1.2,}...{\mathrm{,\; 2}}^N}\right)}_{\overline{\mathrm{A.}}},{\left({{\displaystyle \theta }}_{i=\mathrm{0.1,}...,N}^{k=\mathrm{1.2,}...2^i}\right)}_{\overline{\mathrm{A.}}}\right)$

using a trained data-based network parameter model depending on microstructure configuration parameters ( A. ( x )), with the microstructure configuration parameters ( A. ( x )) the microstructure of the integration point in question ( x ) - Using the Deep Material Network (1) to find a stress and a consistent tangent for each integration point ( x ) to determine; - Performing the mechanical simulation based on the stresses and consistent tangents at the integration points ( x ). Computer program which is set up to carry out all the steps of a method according to one of the Claims 1 until 6th to execute. Electronic storage medium on which a computer program is based Claim 7 is stored.

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