15th World Congress on Computational Mechanics (WCCM-XV)
8th Asian Pacific Congress on Computational Mechanics (APCOM-VIII)
Virtual Congress: 31 July – 5 August 2022
S. Koshizuka (Ed.)

SURROGATE MODEL OF ELASTIC LARGE-DEFORMATIONBEHAVIORS OF COMPLIANT MECHANISM USING
CO-ROTATIONAL BEAM ELEMENTKAI SUTO1 , YUSUKE SAKAI2 , KOTARO TANIMICHI1 AND TAISUKE OHSHIMA1
1Nature Architects, Inc.
Tokyo 107-0052, Japan
info@nature-architects.com
2

Sony Computer Science Laboratories, Inc. Kyoto Laboratory
Kyoto 600-8086, Japan
Yusuke.C.Sakai@sony.com

Key words: Compliant mechanism, Large-deformation analysis, Surrogate model, Co-rotational beam
element
Abstract. We propose a surrogate model for predicting in-plane nonlinear structural deformations ofa compliant mechanism. Our model utilizing a 2-dimensional co-rotational beam element extracts theessential deformation degrees-of-freedoms (DOFs) of bending flexible beams. The total number of DOFs
of nodes at both ends of a 2-dimensional beam is six, while the number of deformation DOFs is three,i.e., axial extension, symmetric bending, and anti-symmetric bending. Therefore, it enables us to reducethe computational cost, from six to three, associated with the models by using the essential deformation
DOFs of the co-rotational beam element. Moreover, it is difficult to predict the nonlinear responses offorces derived from displacements of a compliant mechanism due to bifurcation of the deformation-path.
To overcome the problem, we generate the datasets by applying external forces and use the inverse response for constructing the surrogate models. In the numerical example, large-deformation behaviorsof several types of compliant mechanisms are predicted by our surrogate models constructed by three
typical learning algorithms: polynomial approximation, radial basis function, and neural network. The
prediction performances and computational costs are investigated for verifying that they can be beneficial tools for designing a compliant mechanism with nonlinear elastic deformation behaviors.
1

INTRODUCTION

In a wide range of engineering fields, mechanical design problems crucially affect the total performance of a product. In particular, designing a largely deformable structure is not a simple task due toits geometrical nonlinearities. We traditionally design the structure with desirable deformation DOFs
by rigid-body mechanism, which is composed of rigid links and additional joints such as pin and roller.
While compliant mechanism can realize the desirable motions by the designers by utilizing elastic deformation of each flexible member [1]. Compliant mechanisms have opened new avenues for the design andfabrication of structures with complex deformation DOFs. Following Ref.[1], there are two main advantages of using compliant mechanisms: (i) cost reduction and (ii) increased mechanical performance for
precise deformations. The former advantage leads to fabricating them monolithically. It contributes toreducing the number of additional parts such as rotating and sliding joints, and it is possible to generate1Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshimaa lightweight structure rather than a rigid-body mechanism. The latter leads to reducing the errors anddamages due to backlash and friction of joints. However, there have been some difficulties in broadening
compliant mechanisms to general applications. Design of them is still hard for many designers, because
the sophisticated knowledge and techniques of computational modeling and analysis are required dueto their complex geometrical configurations and specific deformation behaviors. Moreover, numericalsimulation with geometrical nonlinearities can be a time-consuming task.
Pseudo-rigid-body model is often used for describing the large-deformation behaviors of a compliant
mechanism [1]. A basic idea of the pseudo-rigid-body model is based on a simple analyzing system forreducing the deformation DOFs by using a combination of rigid bars and torsional springs, and it is also
associated with the rigid-body mechanism theory. However, it is necessary to construct a computationalmodel based on the geometrical shape of each structure by trial and error.
A data-driven approach can improve the simplicity of the design process of complex structures withhighly nonlinear deformation behaviors [2, 3]. Recent advances in data-driven approaches to design
structures have shown superior effectiveness for predictions of their mechanical properties. There havebeen many developments for a surrogate model predicting elastic large-deformation behaviors of a compliant mechanism. Malomo et al. [2] proposed FlexMaps, which allow us to carry out fast and accurate
numerical simulation for large-deformation of a compliant mechanism. The idea of their method is thatthey utilize the bending beam model, with the stiffness matrix constructed by the datasets of the nonlinear structural responses of a compliant mechanism, as an equivalent model to the original structure.
Papadopoulos et al. [3] predict the nonlinear deformation behaviors of a carbon nanotube by using
the surrogate model, which has the essential deformation DOFs, constructed by the scattered datasetscollected by finite-element simulations. A data-driven approach enables us to propose a method for
obtaining nonlinear deformations of a compliant mechanism without resort to sophisticated techniques
required for designing a complex geometrical configuration.
In this paper, we propose a surrogate model for predicting in-plane nonlinear structural deformation behaviors of a compliant mechanism composed of multiple bending beams. The basic idea of our surrogate
model as briefly illustrated in Fig. 1 is that the use of an equivalent beam model with reduced deformation DOFs overcomes the computational task caused by a complex geometrical configuration and large
DOFs of detail model composed of multiple beams. Our surrogate model is constructed by the following
three steps: (i) preparing a structural component, with essential deformation DOFs, that is appropriate for being incorporated into the numerical simulations based on finite-element analysis, (ii) carryingout finite-element analysis with geometrical nonlinearities for sampling the datasets of static structural
responses of detail model composed of multiple beams, and (iii) learning the datasets for predicting
the nonlinear structural responses of a compliant mechanism. At step (i), we use a 2-dimensional corotational beam element, which enables us to separate the DOFs into rigid and natural straining modes.
By using the co-rotational beam element with essential deformation DOFs corresponding to the natural
straining modes, the computational costs associated with dimensions of datasets can be reduced from six
to three. The datasets collected by numerical simulation at step (ii) are stress-strain relationships derived
from the natural straining modes of a co-rotational beam element. Our surrogate models are based on
three learning algorithms: polynomial approximation, radial basis function, and neural network. In the
numerical example, we investigate the prediction performances of our models and compare the computational costs and deformations with those of the detail models composed of multiple bending beams.

2

Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

Figure 1: Conversion of a compliant mechanism; (a) continuous model, (b) discretized detail model composed ofbending beams incorporated into numerical simulation by finite-element analysis, (c) equivalent model composedof a single co-rotational beam element based on a surrogate model constructed by the scattered datasets of nonlinear
structural responses.

22-DIMENSIONAL CO-ROTATIONAL BEAM ELEMENT

Consider a 2-dimensional bending beam—Euler-Bernoulli beam—composed of two end nodes. The
DOFs on each node of a beam are three: two translations and one rotation; accordingly, the total numberof DOFs is determined as six for a single beam element. To reduce the DOFs, we use a 2-dimensional
co-rotational beam element as explained in Ref. [4]. Figure 2 illustrates the deformation modes of a2-dimensional beam element. By introducing the idea of a co-rotational beam element, the deformation
mode of a beam can be separated into two parts: rigid body mode and natural straining mode. The
rigid body mode is composed of translation and rotation as illustrated in Fig. 2(b). While, the threestrains of the natural straining mode: axial extension, symmetric bending, and anti-symmetric bending,as illustrated in Figs. 2(c)–(e), are represented as v = [u, ϕs , ϕa ]T . The generalized forces corresponding
to v are introduced by t = [N, Ms , Ma ]T .
The equivalent beam model with the nonlinear response between v and t [3] can be implemented into
the computational procedure that we deal with global coordinate displacements pg and forces qg obtainedby carrying out the following two coordinate transformations: (i) between the element and co-rotational
coordinate systems and (ii) between the co-rotational and global coordinate systems.
The transformations from displacements pe and forces qe in the element coordinate system to those in3

Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

Figure 2: Deformation of a 2-dimensional beam element; (a) motion of beam element in global and element
coordinate systems, (b) rigid body mode: translation and rotation, (c) axial extension, (d) symmetric bending, (e)anti-symmetric bending.

the co-rotational coordinate system are formulated as

−1 0
0

0
2/l 


  0


S1
0
−1
1
,
pe = Sv and qe = St with S =
=

S2
0
0 

 1
 0
0 −2/l 
0
1
1


(1)

where S and l denote transformation matrix and the current length of a beam, respectively. We also
introduce the transformation from displacements pg and forces qg in the global coordinate system to peand qe by using a matrix Re as follows:
pg = Re pe = Re Sv and qg = Re qe = Re St,
where Re is expressed as a combined form with respect to the rotation matrix R and written by
cos ρ − sin ρ 0RRe =with R =  sin ρ cos ρ 0  ,
R
0
0
1

4

(2)

(3)

Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

where ρ denotes the rotation between the initial and current axes of a beam. Equations (1)–(3) represent
the transformation between global and co-rotational coordinate systems.
To carry out the gradient-based simulation e.g. Newton-Raphson method, the global coordinate tangent∂tstiffness Kg is derived from the co-rotational tangent stiffness Kd = ∂v
via the coordinate transformations. Note that the element stiffness matrix Ke is divided into the following two terms:
Ke = SKd ST + Kr .

(4)



 r

0 −Q 01
K11 Kr12
Kr =with Kr11 = Kr22 = −Kr12 = −Kr21 = −Q N 0 ,Kr21 Kr22
l
0
0 0

(5)

The second term Kr is computed bywhere Q(= −2Ma /l) represents shear force complementing the equilibrium system in the anti-symmetricbending mode of a co-rotational beam element. The global coordinate tangent stiffness matrix Kg isfinally computed by using the element stiffness matrix Ke and rotational transformation matrix Re as
Kg = Re Ke RTe .
3(6)

GENERATION OF DATASETSIn this section, we explain how to generate the datasets used for training our surrogate models for predicting in-plane deformations of a compliant mechanism. The procedure for generating the datasets isdivided into the following three steps: (i) generate n random samples of t with the Latin hypercube sampling (LHS) method, (ii) carry out large-deformation analyses, corresponding to n sampled datasets of t,
of a compliant mechanism composed of multiple beams under the boundary condition as shown in Fig.
3, and (iii) store the datasets consist of pairs of the co-rotational coordinate strains v and stresses t, which
are derived from the displacements and forces at both ends of a compliant mechanism, respectively, as
follows:




u
N
v =  ϕB − ϕA  andt =  MB − MA  ,
(7)
A
B
A
B
ϕ +ϕ
M +M
where N, M A , and M B denote external force and moments applied to node A and B, respectively, as
illustrated in Fig. 3(a); u, ϕA , and ϕB denote translation at node B and rotations at nodes A and B,
respectively, as illustrated in Fig. 3(b). Note that node A of a detail model has a pin-support and node B
is roller-supported that can move along X-direction.
We apply external forces instead of forced displacements to end nodes of a detail model for samplingdatasets, because multiple deformed equilibrium states can be obtained by bifurcation of the deformationpath, which means that it is unfavorable to construct a surrogate model due to the noise. To overcome theproblem due to bifurcation, the Newton-Raphson method is used for solving the numerical simulation,
because we can eliminate the ’ill’ datasets, which cause the disturbance such as steepness on the wholedatasets, by utilizing the divergence of the simulations.

5Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

Figure 3: Boundary condition given to a detail model for generating datasets; (a) initial state, (b) deformed
equilibrium state.

4

SURROGATE MODELS

Surrogate model is appropriate for expressing approximately a highly complex function of disordered(scattered) datasets, which are difficult to be represented in any analytical form. Constructing a surrogate
model corresponds to finding a feasible set of hyperparameters determined by minimizing the errors between the target and prediction outputs. It is crucial to use a differentiable function as a basis function ofsurrogate model, because its derivatives are required to find a feasible solution for the optimization problem minimizing the errors. We construct three surrogate models: polynomial approximation, radial basis
function (RBF), and neural network (NN), because they are typical learning algorithms for estimating anonlinear function and have been used for solving a wide range of engineering problems. Note that input
and output used for learning of our surrogate models are strains and stresses, respectively.
4.1

Polynomial approximation

The nonlinear response t(v) of a single equivalent beam element is estimated by using polynomial
approximation. Based on a constitutive equation of a co-rotational beam element, the Ξ-dimensional
Taylor expansion of t(v) has the following canonical expression:Ξt (v) = ∑ Kξ vξ ,

(8)

ξ=1

where Kξ ∈ R3×3 Hξ denotes stiffness matrix corresponding to the ξth term. Letting h.o.t stand for higher
order terms, Eq. (8) is rewritten by using the components of vξ asTt (v) =K1 [u, ϕs , ϕa ]T + K2 u2 , ϕs 2 , ϕa 2 , uϕs , uϕa , ϕs , uϕaT+K3 u3 , ϕs 3 , ϕa 3 , u2 ϕs , u2 ϕa , uϕs 2 , uϕa 2 , ϕs 2 ϕa , ϕs ϕa 2 , uϕs ϕa + h.o.t.

(9)The total dimension Ξ of our polynomial approximation is specified as the value, where the best prediction performance is obtained, in the range of 1 ≤ Ξ ≤ 10.

6

Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

4.2

Radial basis function

In a wide range of engineering problems, the RBF has been generally used as an approximation for
estimating a response surface for large scattered datasets [5]. The basic idea of this approach is togenerate the response surface as a linear combination composed of multiplications of weights and RBF.
The obtained response surface passes on the target datasets. The basis function φ(x, c) with respect
to an input variable x is defined as a real function depending on the distance from the RBF center cto x. A generalh form of φ(x,ic) can be described by φ (x, c) = φ (||x − c| |). We use Gaussian function
φ(x, c) = exp −ε (||x − c| |)2 as RBF, where ε is the hyperparameter defined as the ’influence’ of asingle training data on all datasets. Here, RBF centers c are kmax strains v1 , . . . , vkmax sampled from thegeneration of datasets. The interpolation function is written by the following linear combination:
kmax

f (v) = ∑ wk φ (||v − vk | |) ,

(10)

k=1

where wk is the RBF weights on the kth relationship between the input and output. To find a feasible setof the components of weight vector w = [w1 , . . . , wkmax ], the interpolation function satisfies the conditionf (vk ) = tk for the kth dataset. We finally obtain the RBF weights w by carrying out least squared method
for minimizing the differences || f (vk ) − t|| for all k.
4.3

Neural network

NN, an artificial network model inspired by a combination of biological neurons [6], can be utilized asa surrogate model for predicting nonlinear responses of large-deformation behaviors of an elastic beammodel [3]. We use multilayer perception (MLP), which is a general form of NNs and consists of fully
connected multiple layers composed of units. The network constructed in this paper has a single hidden
layer following a classical MLP, and the hidden layer has 50 units, as shown in Fig. 4. The units in the

h1
h2
h3
h4

u

N

s

Ms

a

Ma

Input Layer ∈3h48

Output Layer ∈

h49

3

h50

Hidden Layer ∈50

Figure 4: Architecture of our NN based on a classical MLP with one hidden layer composed of 50 units.

7Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

hidden layer denoted by hm ; m = 1, . . . , 50 are computed as3

hm = σ

∑

!
(1)
(1)
wml vl + bm

,

(11)

l=1
(1)

(1)

where σ(·), wml , and bm denote activation function, weight parameters, and bias parameters, respectively. Superscript (1) in Eq. (11) indicates that the parameters consist of the first layer of our NN.
We choose the logistic sigmoid function 1/(1 + e−v ), which is differentiable, as σ(·) for all connections
between the units in two neighboring layers. The units in the output layer are written by!
!
50tl (v, w, b) = σ

3

(2)

(1)

(1)

∑ wlm σ ∑ wml vl + bm

m=1

(2)

+ bl

,

(12)

l=1

where w and b represent vectors composed of weight and bias parameters, respectively.
The learning process of NN is classified into two phases: the training and test phases. We randomly
assign 90% of datasets to the training phase and 10% to the test one. During the training phase, the loss
max
function defined as mean squared error ∑kk=1||tk − t̂k ||2 /kmax is minimized based on error backpropagation, which is a computational procedure to obtain the partial derivatives of predictions with respect tovariables from outputs to inputs. kmax is the number of datasets collected by large-deformation analysis
with the detail model composed of multiple beam elements, and tk and t̂k represent the kth target andprediction outputs, respectively. We choose the batch size equal to the total number of datasets; batchlearning is performed. Optimization is therefore carried out via gradient descent. Learning rate is 0.0001.
Training our NN is performed until 500000 epochs. The set of hyperparameters is determined manuallyby trial and error. We use PyTorch [7], a Python library of NN, for constructing the architecture of our
NN and carrying out the learning process.
5

RESULTSTo investigate the prediction performances of the three surrogate models as introduced in Sec. 4,
we carry out the training and test of scattered datasets for nonlinear responses between strains v andstresses t of four types of compliant mechanisms: Simple beam, Kazaguruma, Spiral, and Rhombus.
The numbers of datasets for Simple beam, Kazaguruma, Spiral, and Rhombus are 1000, 1000, 487, and1000, respectively. Note that we eliminate the datasets not converged in the numerical simulations withthe detail model of Spiral.
By carrying out standardization, a statistical procedure to tune the variables of datasets, the mean isset as 0 and standard deviation 1 for each variable. To eliminate the deterioration of the performances
due to the different scales and types of variables, standardization is to be carried out prior to the learning
process. Computation is carried out on a PC with Intel Core i7-9750H 4.50 GHz, 32.0GB RAM, and six
cores.
5.1

Comparison of our surrogate model predictions

In the following examples, we use a material with Young’s modulus 300 MPa and Poisson’s ratio 0.3.
The beam has a rectangle section with width 2 mm and height 4 mm. The initial length of each beam
element is 100 mm. We investigate the following average differences between the numerical simulation
results with detail models and the surrogate model predictions: (i) distance of vectors t = [N, Ms , Ma ]T =8

Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

(a)

(b)

(c)

(d)

Figure 5: Geometrical configurations of compliant mechanisms used for investigation of prediction performancesof our surrogate models; (a) Simple beam, (b) Kazaguruma, (c) Spiral, (d) Rhombus.

[t1 ,t2 ,t3 ]T and t̂ = [N̂, M̂s , M̂a ]T = [tˆ1 , tˆ2 , tˆ3 ]T , (ii) axial stresses, (iii) symmetric bending stresses, and (iv)
anti-symmetric bending stresses, which are defined as
1kmax

1 kmax |tlk − tˆlk |||tk − t̂k ||× 100 (%) and∑ |tlk | × 100 (%)||tk ||
kmax k=1
k=1

kmax ∑(13)

with l ∈ {1, 2, 3}.
Table 1 shows the average differences of forces at both ends of the detail model and the equivalent beammodel based on our surrogate models. For all compliant mechanisms, the polynomial approximation and
NN are not performed better than our RBF. Our trained RBF can produce good prediction performancesof the numerical simulation with the detail models. The average differences obtained from our RBF for
all compliant mechanisms are lower than 1%. Notably, the prediction performances of RBF constructed
for Rhombus are good, and the average differences of the force vector and its components are lower than
1.0 × 10−3 %. As seen in the results, the in-plane nonlinear structural responses of the four compliant
mechanisms can be predicted practically by our RBF.
5.2

Comparison of computational time of detail and surrogate models

We compare the computational time of numerical simulations with the detail model and prediction
with our RBF. Three compliant mechanisms, whose initial shapes are straight and composed of single,
five, and ten Spirals, are set to serve as our benchmark. We set ten random cases of boundary conditions
applied to both ends of the compliant mechanisms, and measure the average computational time required
for a convergence of a numerical simulation.
The average times of the detail model and our RBF are 116.2 ms and 7.5 ms for single, 7278.2 ms and45.7 ms for five, 86358 ms and 227.4 ms for ten Spirals, respectively. The large-deformation analyses for9Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

Table 1: Average differences between the numerical simulation results with detail models and our surrogate model
predictions for nonlinear structural responses of four types of compliant mechanisms.

(unit: %)Simple beam

Kazaguruma

RhombusSpiral

t
N
Ms
Ma
t
N
Ms
Ma
t
N
Ms
Ma
t
N
Ms
Ma

Polynomial approximation
1.515010.5530.80860.12810.79940.25620.95723.32460.00070.00650.00120.00050.2684
0.10900.80050.3026

RBF
0.00220.05050.00090.00020.00100.00670.00120.00290.000030.00020.000030.00020.00030.00020.00100.0004NN
0.18912.05640.4382
1.13810.1483
0.23650.38470.70260.01240.03800.05750.02630.13110.10500.26360.2405

single, five, and ten Spirals are approximately 16, 160, and 380 times faster using the equivalent beammodel based on our trained RBF than the detail model, respectively.
5.3

Prediction of in-plane deformation behavior of compliant mechanisms composed of multiple
Spirals

We further validate the proposed surrogate model in two numerical examples: Straight model andGrid model. Our RBF trained with a single Spiral is implemented for predicting in-plane deformation
behaviors of the examples.
Figure 6 shows the initial and deformed shapes of the detail models composed of multiple beam members and the equivalent beam models based on our RBF for predicting the deformation behaviors of Spiral. The geometrical configurations of the detail models are illustrated by black piecewise linear curves,and the equivalent beam models blue lines. Straight model, which is unstressed in the initial state, isarranged along x-axis. The detail model has five Spirals, pin-supported at one end, and roller-supported
at the other end allowing translational displacement in x-direction. The equivalent beam model has six
nodes, and the boundary conditions are equivalent to the detail one. We apply external forces of −0.0033
N along the axial direction and symmetric and anti-symmetric moments of 0.21107 Nmm and 0.026528
Nmm at both ends of the model, respectively.
Grid model has 200 × 200 (mm) square boundary and four internal Spirals intersecting at the center.
The detail model is composed of twelve Spirals, and the equivalent beam model has twelve co-rotational
beam elements arranged in the grid. The three nodes in the left column are fixed, and the others canmove in xy-plane. Forced displacements of 100 mm are applied at the node connecting the two Spirals
in the right column towards the opposite direction of the center node of the grid.
As seen in Figure 6(b), we can see that the coordinates of all nodes between the detail and equiv-

10Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

y

y
x

x

(a)

(b)

y

y

x

x

(c)

(d)

Figure 6: Initial and deformed shapes of compliant mechanisms with multiple Spirals. (Grey) Detail model. (Blue)
Equivalent beam element based on our RBF; (a) initial shape of Straight model, (b) deformed shape of Straightmodel, (c) initial shape of Grid model, (d) deformed shape of Grid model.

alent beam models of Straight model are almost the same. The average and maximum differences in
x-direction are 0.0056 mm and 0.0099 mm, respectively, and those in y-direction are 0.0026 mm and0.0059 mm, respectively. The differences are sufficiently small for the total length of Straight model.
Also, Fig. 6(d) shows that the coordinates of all nodes between the detail and surrogate models of Gridmodel are very close. The average and maximum differences in x-direction are 0.1055 mm and 0.7906
mm, respectively, and those in y-direction are 0.0844 mm and 0.3710 mm, respectively. Although thedifferences for Grid model are larger than those for Straight model, they are small for practical use of the
proposed model for finding better candidate solutions.
6

CONCLUSIONS

The main purpose of this paper is to propose a surrogate model for predicting nonlinear deformation
behaviors of a compliant mechanism. We use a 2-dimensional co-rotational beam element, with thethree essential deformation DOFs, for reducing computational costs of large-deformation analysis of adetail model composed of bending beams, with six DOFs at each end. This procedure can degenerate thedimensions of inputs and outputs, from six to three, used for training our surrogate models. The datasets
used for constructing the surrogate models are firstly sampled as forces and displacements by carryingout the numerical simulation by finite-element method with the detail model, and they are converted into
stresses and strains associated with the three essential deformation DOFs in the co-rotational coordinate
system. We adopt the LHS method for generating random sets of forces applied to both ends of the detailmodel. Note that the forces and displacements at both ends of the detail model correspond to inputs andoutputs for the sampling datasets, respectively; however, the co-rotational coordinate strains and stresses
correspond to inputs and outputs for constructing a surrogate model, respectively. In the numericalexample, we investigate the prediction performances of the constructed surrogate models for the four11Kai Suto, Yusuke Sakai, Kotaro Tanimichi and Taisuke Ohshima

types of compliant mechanisms: Simple beam, Kazaguruma, Spiral, and Rhombus. Comparing thedifferences of the target and prediction outputs, the results show that our RBF exhibits better prediction
performances than our polynomial approximation and NN. Notably, our RBF has great performancefor predicting in-plane deformation behaviors of Spiral and Rhombus. To validate that our surrogate
models enable us to resolve high computational costs for carrying out large-deformation analysis withthe geometrical nonlinearities, we measure the computational time of the detail model and our RBF
for the different sizes of Spirals. As a result, the use of our RBF for Spiral can significantly reduce
computational costs. Furthermore, the average differences of coordinates of all nodes between the detailmodel and our RBF for Straight and Grid models are sufficiently small. The results suggest that oursurrogate model can be an alternative solver for obtaining candidate in-plane deformation behaviors of acompliant mechanism.
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