The accurate, stable, and robust simulation of or materials typically used in soft robotics relies crucially on the accuracy of the parameters used in the simulation. For skin simulations, for example, hyperelastic material models are often used due to their ability to approximate the behavior of elastomeric materials, such as silicone and urethane, for instance.
Traditionally, characterization of elastomers is done by testing the uniaxial and biaxial, and sometimes triaxial (i.e., volumetric) behavior of the material. Material parameters are then fitted to the resulting data using analytical models that assume a particular deformation mode in the sample. However, multiple tests are typically required for good fits, making such traditional solutions time consuming because biaxial, and particularly triaxial setups are relatively complex and costly. Moreover, because the stiffness of a material depends on the resolution and order of finite elements that are used for the simulation, reliance on analytical material models for parameter estimation can lead to inaccurate predictions of material characteristics, such as elasticity, or, more generally, the deformation of an object and the corresponding stresses and strains under a specified load. Consequently, there is a need in the art for a material simulation solution that is fast, cost effective, and accurately describes one or more characteristics of the material being simulated.
There are provided systems and methods for simulation-based material characterization, substantially as shown in and/or described in connection with at least one of the figures, and as set forth more completely in the claims.
The following description contains specific information pertaining to implementations in the present disclosure. One skilled in the art will recognize that the present disclosure may be implemented in a manner different from that specifically discussed herein. The drawings in the present application and their accompanying detailed description are directed to merely exemplary implementations. Unless noted otherwise, like or corresponding elements among the figures may be indicated by like or corresponding reference numerals. Moreover, the drawings and illustrations in the present application are generally not to scale, and are not intended to correspond to actual relative dimensions.
As further shown in
By way of overview, it is noted that in engineering, it may be advantageous or desirable to characterize materials in order to be able to predict their behavior in simulations. One notable advantage of the novel and inventive technique disclosed in the present application is that material characteristics are directly characterized in the context of simulation. This has the following advantages: (1) Higher accuracy of estimated parameters because we directly take a simulation representations into account. (2) Although conventional analytical model fitting assumes uniaxial, biaxial, and triaxial behavior to be decoupled, with simulation representations of test specimens we can advantageously couple these behaviors and can achieve higher prediction accuracy with fewer mechanical tests.
The material characterization performed by the material simulation systems and according to the methods disclosed herein may be used in the design of objects made of the characterized material, but assuming a variety of different geometries. In other words, the material characterizations disclosed in the present application advantageously enable the simulation of objects having arbitrary shapes based on a characterization performed using a small test sample of the material.
It is noted that although manufacturing system 160 is depicted as distinct from material simulation system 100, that representation is merely exemplary. In other implementations, manufacturing system 160 may be included as a component of material simulation system 100, and may be integrated with material simulation system 100, or may be remote from but communicatively coupled to material simulation system 100. That is to say, in some implementations, manufacturing system 160 may be under the control of computing platform 102.
It is further noted that, in various implementations, one or more material characteristics 158, when simulated using differentiable material simulation software code 110, may be stored in system memory 106 and/or may be copied to non-volatile storage (not shown in
Although user system 130 is shown as a desktop computer in
It is also noted that although the present application refers to differentiable material simulation software code 110 as being stored in system memory 106 for conceptual clarity, more generally, system memory 106 may take the form of any computer-readable non-transitory storage medium. The expression “computer-readable non-transitory storage medium,” as used in the present application, refers to any medium, excluding a carrier wave or other transitory signal that provides instructions to hardware processor 104 of computing platform 102, or to a hardware processor of user system 130 (identified as hardware processor 134 below by reference to
Moreover, although
In one implementation, for example, computing platform 102 of material simulation system 100 may correspond to one or more web servers, accessible over a packet-switched network such as the Internet, for example. Alternatively, computing platform 102 may correspond to one or more computer servers supporting a wide area network (WAN), a local area network (LAN), or included in another type of limited distribution or private network.
The implementation shown in
Referring first to uniaxial testing apparatus 240a, uniaxial testing apparatus 240a is configured to stretch material 250 in a single direction, e.g., in the “x” direction. As shown in
Analogously, biaxial testing apparatus 240b is configured to stretch material 250 in perpendicular directions, e.g., in the “x” direction and in the orthogonal “y” direction. As further shown in
In addition, and as also shown in
It is noted that in conventional analytical model fitting, only the mid-part of the test specimen is considered. This mid-part is uniformly stretched in one direction for a uniaxial test, and in two directions for the biaxial test. In all other directions, the stress is zero. However, according to present novel and inventive solution, the boundary conditions are modeled in simulation, and the simulated test specimen undergoes non-uniform stretching. Because the effects at the boundaries are taken into account in the present approach, we can perform less mechanical testing while increasing the accuracy of the estimated model parameters.
Uniaxial testing apparatus 340 corresponds in general to uniaxial testing apparatus 240a, in
By way of overview of the material simulation solution disclosed by the present application, and referring to the exemplary physical testing and corresponding simulation depicted in
Representing material 250/350 with a non-analytical parameterized model, such as a finite element discretization, for example, the hyperelastic material parameters that minimize differences between simulated and measured displacements d 386 and
for every moving end, and force-displacement material.
To improve robustness to noise, the forces f 384 applied in simulation 380 are treated as parameters, and the displacement objectives and force objectives of the form
weighted by wf are jointly optimized.
Summing up displacement and force objectives for every moving interface k, and every material sample i, the parameters p that minimize the following characterization objective are sought:
To minimize this objective, we express standard finite element degrees of freedom that are rigidly moving with the load cells and carriages shown in
where we ask external forces fext to be in balance with the internal response fint of the sample of material 250/350. In order to keep material parameters within physically feasible ranges, they are bound from above and below where necessary. It is noted that this formulation enables the combined characterization from different tests (e.g., from uniaxial and biaxial test results), assigning test samples of material 250/350 having different dimensions the same material parameters p.
The functionality of systems 100 and 130 including differentiable material simulation software code 110 will be further described by reference to
Referring to
In some implementations, testing apparatus 140/240a/240b/340 may operate independently of system 100 or user system 130, in which use cases obtaining result 142/342 of the physical test performed on material 250/350 may correspond to simply receiving result 142/342 from testing apparatus 140/240a/240b/340. However, in other implementations, as noted above, testing apparatus 140/240a/240b/340 may be a component of system 100, or may be controlled by user system 130. In those implementations, obtaining result 142/342 in action 491 may include executing differentiable material simulation software code 110 to control testing apparatus 140/240a/240b/340 to perform the physical test on material 250/350.
As described above by reference to
Flowchart 490 continues with selecting parameterized model 382 of material 250/350 based on obtained result 142/342 (action 492). The selection of parameterized model 382 of material 250/350 based on obtained result 142/342 may be performed by differentiable material simulation software code 110, executed by hardware processor 104 or 134. In some implementations, parameterized model 382 may be an existing model usable as is, while in other implementations parameterized model 382 may be an existing model that is customized for material 250/350. In yet other implementations, parameterized model 382 may be developed specifically for material 250/350.
Parameterized model 382 of material 250/350 may include a differentiable mathematical representation of material 250/350, such as a differentiable finite element representation of material 250/350. In the interests of conceptual clarity, the discussion below first describes how to compute analytical gradients of a single sample of material 250/350 tested on uniaxial test apparatus 140/240a/340, and then provides a roadmap for making a finite element representation differentiable.
Referring to
To disclose numerical optimization of the characterization problem expressed as Equation 1 above, it is sufficient to study the single sample case. For the use case in which a single sample of material 250/350 undergoes a physical test, as shown in
In this simplified form of the objective expressed by Equation 1, we collect the unknown optimization variables in a vector y=(p, f), and the elastic response of the model in a vector z=(x, d) as this aids in keeping the notation concise.
For numerical optimization, an analytical gradient is required. Due to the implicit dependence of the elastic response on the unknowns, the gradient is the total derivative:
d
y
g
char=∂ygchar+∂zgchardyz. (Equation 3)
Most entries of the two partial derivatives, ∂ygchar=(0T, f−
f(y,z(y))−fint(p,z(y))−(0T,f)=0T (Equation 4)
that balances the nonlinear internal forces with the applied force. Because an elastic response that fulfills this constraint for sets of parameters and forces can be found in a neighborhood of a given y, the constraint can be considered constant, and its derivative to be zero:
d
y
f=∂
y
f+∂
z
fd
y
z=0T. (Equation 5)
Flowchart 490 continues with performing a simulation of the physical test performed on material 250/350 by testing apparatus 140/240a/240b/340, using parameterized model 382 of material 250/350 to generate simulated result 386 (action 493), and performing a comparison of simulated result 386 with obtained result 142/342 (action 494). Flowchart 490 further continues with adjusting parameter values of parameterized model 382, based on the comparison performed in action 494, to improve simulated result 386 (action 495), followed by predicting one or more characteristics 158 of material 250/350 based on the adjusted model parameters (action 496). For example, where the material characteristic being modeled is the elastic response of a material, action 495 results in a parameterized model and corresponding material parameters that make it possible to predict the elastic response of the material in simulation, at any point in time. Actions 493, 494, 495, and 496 may be performed by differentiable material simulation software code 110, executed by hardware processor 104 or 134.
The application of the implicit function theorem discussed above by reference to action 492 provides a recipe to compute analytical gradients: Whenever the set of unknowns y is updated, a simulation is performed to find the equilibrium z(y) for which the internal and external forces are in balance. That is to say simulated result 386 is compared to obtained result 142/342 under a constraint that ensures that applied forces are in equilibrium with the elastic response of material 250/350 used as the test sample. The derivative of the elastic response can then be computed by solving the system of equations dyz=−(∂zf)−1∂yf, and Equation 3 above can be evaluated. For the multi-interface, multi-sample case, the simulations may be performed for every sample i of material 250/350, taking into account all forces k that act concurrently.
Because internal forces do not directly depend on f, the partial derivative ∂yf of the equilibrium constraint can be computed by forming the derivative ∂zfint, and subtracting the constant matrix with a 1 in the last row and column. The partial derivative ∂zf is the non-constant tangent stiffness or stiffness matrix ∂zfint, which can be computed from the standard matrix by applying the chain rule.
This formulation can be used to fit common hyperelastic material models to acquired displacement-force curves. While the technique is applicable to any model for which a strain energy density Ψ exists, in the interests of conceptual clarity, the present approach is described by reference to three representative hyperelastic materials that are commonly used for elastomer simulation, and are available in commercial packages: the Neo-Hookean model, a generalized Mooney-Rivlin model, and the 3rd-order Yeoh model.
Conditioned on having a simulator that enforces incompressibility with constraints, the present method could be used to fit both compressible and incompressible models. However, while elastomers are commonly considered incompressible or nearly incompressible, finite element implementations often assume a compressible model because constraint-based approaches tend to increase the time and implementational complexity, or can cause locking, as known in the art. Consequently, the present approach utilizes a compressible model to increase computational efficiency and to reduce cost.
For compressible models, it is important to fit parameters that do not lead to simulation instabilities. To ensure stability, material 250/350 may be parameterized using the consistency with linear elasticity, setting the Poisson's ratio ν to a value that is sufficiently far from 0.5. The remaining parameters may then be fitted.
Volume preservation terms that depend on the determinant of the deformation gradient, J=detF, may be exponentiated with an even number to get rid of the sign. Hence, it is important to keep corresponding parameters from taking on negative values. Negative values for parameters that weigh terms that depend on the first or second invariants, I1 or I2, can undesirably lead to negative energies in moderately deformed material 250/350 of poor quality, or highly-stretched material 250/350 of high quality. Because negative energies are non-physical, it may be advantageous or desirable to bound these parameters from below, to keep them non-negative.
To demonstrate the present method, the following compressible versions of the Neo-Hookean model are used:
as well as a generalized Mooney-Rivlin model:
ΨMR=C10(Ī1−3)+C01(Ī2−3)+C11(Ī1−3)(Ī2−2)+D1(J−1)2,
and the 3rd-order Yeoh model:
ΨY=C10(Ī1−3)+C20(Ī1−3)2+C30(Ī1−3)3+D1(J−1)2+D2(J−1)4+D3(J−1)6.
Referring to
To be compatible with a standard finite element representation, the coupling of a subset of degrees of freedom to rigidly moving parts and the differentiation of internal forces with respect to parameters requires further discussion. While the characterization described above is independent of the element type and its order, it may be advantageous or desirable to represent material 250/350 being tested as being composed of equally-sized hexahedral elements. The reason for this is two-fold: (1) numerical integration using standard Gauss quadrature is more accurate for hexahedral than for tetrahedral elements, and (2) there is no distortion in the mapping from real to natural coordinates for cubical elements. Consequently, by using cubical and equally-sized hexahedral elements, the resolution dependence and order dependence of fitted parameters can be studied, thereby advantageously avoiding any bias due to elements of different shape and size.
To run coupled simulations, evaluations of the internal forces fint, and the tangent stiffness ∂zfint are necessary. To evaluate the undeformed or deformed configuration at the neutral coordinates ξ∈3 within an element, the standard Lagrange shape functions Ni(ξ) corresponding to nodes i of the element are relied upon. For example, for the undeformed configuration, we interpolate the undeformed nodes Xi∈
3 as:
X(ξ)=Σi=1nXiNi(ξ). (Equation 6)
To measure strains within an n-node element, the deformation gradient is defined as the product of the Jacobian of the interpolated deformed nodes xi, and the inverse of the Jacobian Xξ of the undeformed configuration:
F(ξ)=(Σi=1nxi∂ξNi(ξ))Xξ−1(ξ) (Equation 7)
where the partial derivatives ∂ξ of the shape functions are, in general, not constant.
Applying external forces or prescribing displacements, energy is stored within the discretized solid model of material 250/350. To determine this internal energy, the strain energy density Ψ of the hyperelastic material is integrated over the volume ε of the isoparametric element, taking the change of variables from real to natural coordinates into account:
E
int(
As indicated, the internal energy depends on the deformed nodes of the discretized mesh model, collected in a vector
In order to evaluate the energy, the integral may be approximated over ε with an m-point Gauss quadrature:
Σj=1mwjΨ(F(
where wj is the weight that corresponds to point ξj. Softer elastomers such as soft silicones tend to sag significantly under gravity, especially in the low-strain range. As a result, depending on material 250/350, it may be important to account for the work done by gravity. To this end, the dot product can be integrated between the gravitational vector g and the interpolated displacement u(ξ)=Σi=1n(xi−Xi)Ni(ξ) over the volume enclosed by the solid:
E
grav(
approximating the integral with the same quadrature scheme.
Another source of external energy is work done by nodal forces fi∈3:
E
ext(
To compute the deformed configuration, the total potential energy:
E(
is minimized to first-order optimality, ∂xE=0, using a standard Newton.
As discussed above, in an energy-based formulation, the internal energy Eint(
To couple deformed nodes x on the bonding interface to the constrained displacement d along a coordinate axis, the standard degrees of freedom can be assumed to be split into noninterface and interface nodes,
This mapping enables the evaluation of internal forces and the tangent stiffness matrix of the coupled problem with standard quantities:
f
int=∂
∂zfint=(∂z
With the coupling described above, we can correctly predict the non-uniform response of material 250/350 that integrates to the force value f.
To compute the derivative ∂pfint, symbolic differentiation can be used to evaluate the per-element Jacobians of the internal energy Eint with respect to incident nodes, and material parameters. Similarly to the assembly of the tangent stiffness matrix, we assemble elemental contributions to the Jacobian ∂p
In some implementations, the present method may conclude with action 496, described above. However, and although not shown by
Object 170 manufactured based on one or more characteristics 158 of material 250/350 may take a variety of forms. For example, in one use case, object 170 may be an artificial skin for use in medicine, such as to promote healing in burn victims or victims of other traumatic injury. In other implementations, object 170 may be a surface covering or skin for use in manufacture of a robot or other type of machine. In those latter implementations, one or more internal components of the robot or other type of machine may also be manufactured or selected based on one or more characteristics 158.
For example, referring, to
Object 670 corresponds in general to object 170, in
As indicated in
In some implementations, it may advantageous or desirable to render one or more characteristics 158 of material 250/350/650 on display 108 of material simulation system 100 for review by a system administrator, or to render one or more characteristics 158 on display 138 for review by user 131. In those implementations, hardware processor 104 or hardware processor 134 may execute differentiable material simulation software code 110 to render one or more characteristics 158 of material 250/350/650 on respective display 108 or display 138.
However, it is noted that, in some implementations, hardware processor 104 or 134 may execute differentiable material simulation software code 110 to perform actions 491, 492, 493, 494, 495, and 496 (hereinafter “actions 491-496”), as well as subsequent action 497 in an automated process. It is further noted that, as used in the present application, the term “automated” refers to systems and processes that do not require the participation of a human user, such as a human designer or engineer. Although, as noted above, in some implementations, a human designer or engineer, such as user 131, may review the performance of the automated systems and automated methods described herein, that human involvement is optional. Thus, in some implementations, the methods described in the present application may be performed under the control of hardware processing components of the disclosed systems.
Thus, the present application discloses systems and methods for simulating material characteristics that improve on the state of the conventional art. From the above description it is manifest that various techniques can be used for implementing the concepts described in the present application without departing from the scope of those concepts. Moreover, while the concepts have been described with specific reference to certain implementations, a person of ordinary skill in the art would recognize that changes can be made in form and detail without departing from the scope of those concepts. As such, the described implementations are to be considered in all respects as illustrative and not restrictive. It should also be understood that the present application is not limited to the particular implementations described herein, but many rearrangements, modifications, and substitutions are possible without departing from the scope of the present disclosure.