A Flexible Simulation Metamodel for Exploring Multiple Design Spaces

Abstract

We present an approach to build a flexible simulation metamodel whose inputs are generalised, such that the model can be carried forward from one design space to the next.

In fields such as aero-astro and automotive engineering, metamodels are used to substitute complex and computationally demanding simulation with mathematical models, that are simpler and much faster to compute. Metamodels are also receiving attention from the architectural design community, because they can be used to facilitate faster evaluation and/or optimisation time-cycles. However, we argue that typical metamodels lack flexibility when exploring multiple design spaces, because a fresh metamodel has to be recomputed each time new design variables are introduced. Thus, information learned about a design space is not carried forward from one metamodel to the next.

In this paper we present an approach to build a generalised metamodel whose inputs are (i) independent from the design variables, and (ii) are critical to the calculation of the simulation response. Our approach can be used to generalise any type of metamodel, however, in this research, we build up on the Bayesian Network metamodel (BNM), that was introduced in our previous work (Xuereb Conti and Kaijima [1]). The BNM is a knowledge-oriented metamodel that allows bi-directional exploration of relationships between design and engineering response variables. We demonstrate our approach by building a BNM for the finite element analysis of a 2D truss using beam elements, and carry it forward through two subsequent truss designs. Our research shows that the generalised BNM predicts confidently when introduced with new design problems.

Full text

Proceedings of the IASS Symposium 2018
Creativity in Structural Design
July 16-20, 2018, MIT, Boston, USA
Caitlin Mueller, Sigrid Adriaenssens (eds.)
Copyright © 2018 by Zack XUEREB CONTI, Sawako KAIJIMA
Published by the International Association for Shell and Spatial Structures (IASS) with permission.
A Flexible Simulation Metamodel for Exploring Multiple Design
Spaces
Zack XUEREB CONTI*, Sawako KAIJIMAa
*Singapore University of Technology and Design
8, Somapah Rd, 487372
xuereb_zack@mymail.sutd.edu.sg
a Harvard University Graduate School of Design
Abstract
We present an approach to build a flexible simulation metamodel that can input data from different
project sources, implying that the same model can be carried forward from one design space to the next.
In fields such as aerospace and automotive engineering, metamodels are used to substitute complex and
computationally demanding simulation with mathematical models, that are simpler and much faster to
compute. Metamodels are also receiving attention from the architectural design community, because
they can be used to facilitate faster evaluation and/or optimisation time-cycles. However, we argue that
typical metamodels lack flexibility when exploring multiple design spaces, because a fresh metamodel
has to be recomputed each time new design variables are introduced. Thus, information learned about a
design space is not carried forward from one metamodel to the next.
In this paper we present an approach to build a generalised metamodel whose inputs are (i) independent
from the design variables, and (ii) are critical to the calculation of the simulation response. Our approach
can be used to generalise any type of metamodel however, in this research we build up on the Bayesian
Network metamodel (BNM), that was introduced in our previous work (Xuereb Conti and Kaijima [1]).
The BNM is a knowledge-oriented metamodel that allows bi-directional exploration of relationships
between design and engineering response variables. We demonstrate our approach by building a BNM
for the finite element analysis of a 2D truss using beam elements, and carry it forward through two
subsequent truss designs. Our research shows that the generalised BNM predicts confidently when
introduced with new design problems.
Keywords: metamodel, statistics, probabilistic, finite element analysis, Bayesian networks
1. Introduction
Statistical techniques have been used for decades, in fields such as mechanical engineering, to substitute
complex and computationally demanding simulation with a mathematical model, referred to as a
metamodel, that is simpler to interpret and far more computationally efficient to execute. Metamodels
are gaining increasing attention from the architectural design community because they can drastically
reduce evaluation and/or optimisation time-cycles during the early stages of design. In previous work
we proposed a knowledge-oriented metamodel in the form of a Bayesian Network that focuses on
understanding the influence between design variables and simulation response. More specifically, a
Bayesian Network metamodel (BNM) is a type of statistical model that does not distinguish between
independent (!) and dependent (") variables and thus, enables bi-directional exploration between inputs
and outputs.
In this paper we challenge the flexibility of metamodels for architectural design. We highlight that
metamodels do not generalise for new design variables and thus, can hinder the exploratory nature of
the conceptual stages. In other words, each time new design variables are introduced to the design-

Proceedings of the IASS Symposium 2018
Creativity in Structural Design
2
analysis system, a completely new metamodel is built from fresh simulation data. Thus, information
learned about a design space is not carried forward from one metamodel to the next.
In response, we propose an approach to build a flexible metamodel whose inputs can generalise for new
design problems. We argue that since the domain of any mathematical model is bound by the inputs and
output/s that characterise it, the ability for a metamodel to generalise for new problems is directly related
to how generalised the selected inputs are. If we were to look underneath the hood of any engineering
simulation tool, we would observe that inputs to the numerical analysis are not design problem
dependent, but are generalised such that any simulation analysis model is described by one set of
fundamental variables that are derived from domain-related theory and that are critical to calculating the
response. Furthermore, these variables are computationally inexpensive to extract from any simulation
analysis model. For example, in the finite element analysis (FEA) of different parametric truss models,
the moment of inertia, mass, centre of gravity, axial and bending member stiffness, are crucial for
calculating the stiffness matrix, while independent of the parametric description of each truss. While, it
is not our scope to delve into the math underlying the numerical model, we hypothesise that if we can
identify the set of variables that are critical to the math itself through domain expert help, and introduce
them as input variables into the metamodel, we can build a flexible metamodel that can describe physical
behaviour, independent of the variables that describe the design problem, thus can be carried forward
from one design space to the next. Our approach can be adopted for any metamodel, however, in this
paper we will retain our focus on knowledge-based metamodels, introduced in previous work (Xuereb
Conti and Kaijima [1]).
The document will proceed as follows: in section 2 we briefly reintroduce the BNM, in section 3 we
explain and discuss our approach in more detail, using truss design and finite element analysis (FEA) as
an example, and in section 4 we utilise the generalised BNM to explore new variables to demonstrate
the flexibility of our metamodeling approach.
2. Metamodels
2.1. Typical metamodels
A metamodel can be described as a model of a model and is typically expressed as # $ %& ' ( ) ' ,
where & is the numerical model underlying the simulation, and ) is the compressed approximation of &.
Subsequently, ) is used to predict # more efficiently. The most common metamodeling techniques for
approximating simulation models include polynomial regression (Kleijnen [2]), response surfaces
(Kleijnen and Sargent [3]), Kriging (Ankenman, et al. [4]), and Neural Networks (Fonseca, et al. [5]),
among others. The benefits of metamodels for faster design-analysis-optimisation cycles, have attracted
the attention of the architectural design community. For example, Capozzoli, et al. [6] formulate a
metamodel using regression analysis to substitute complex energy calculations and computationally
demanding energy simulation, respectively; Klemm, et al. [7] present a metamodel derived by
polynomial regression from CFD simulation results to derive objective functions for faster optimisation
of building aerodynamics, Tresidder, et al. [8] use Kriging metamodels to optimise CO2 emissions and
construction costs of buildings, and more recently, Wortmann, et al. [9] demonstrate the advantageous
application of metamodel-based optimisation using radial-basis functions, for architectural daylight
optimisation problems.
Typical metamodels can be extremely efficient for searching a design space with a specific objective.
However, the objective is not always clear to the designer and hence why in our previous work we move
beyond the ‘performance-optimisation-driven-design’ agenda, towards a knowledge-based approach
that allows us to make deeper inferences about what is driving the physical behaviour. We argue that a
knowledge-based approach allows for human intelligence to intervene and thus enables us to drive and
control design with our creative intuition. More specifically, we presented a Bayesian Network
Metamodel, which is a type of model that does not compress relationships into a deterministic function
() ' ) unlike typical metamodels, but one that takes into account all sampled input values and
simulation response values, as a joint probability distribution (JPD) that can be accessed and explored
using statistical inference techniques.

Proceedings of the IASS Symposium 2018
Creativity in Structural Design
3
2.2. Knowledge-based metamodels: Bayesian network metamodel (BNM)
A BNM is a type of statistical model that is represented in the form of a directed mathematical graph,
where inputs and outputs are represented as nodes, while relationships between variables are represented
as edges, whose direction indicates a causal influence as shown in Figure 1, B (Pearl [10]). Each node
is encoded with a marginal probability distribution, while each edge is encoded with a conditional
probability distribution matrix. Together, this information is jointly represented as a joint probability
distribution (JPD), that can be accessed and explored using a statistical method called Bayesian
inference. More specifically, a JPD does not distinguish between independent (!) and dependent (")
variables, thus allows bi-directional inference, such that designers and engineers can predict simulation
response for a set of inputs (Figure 1, B1), and/or vice versa, predict the input probability distributions
(PD) for a target response of interest (Figure 1, B2).
Figure 1: Shift from typical forward metamodel (A), to a bi-directional metamodel (B).
3. A Flexible Bayesian network metamodel (BNM)
The use of metamodels for engineering simulation such as FEA, first appeared in fields such as
automotive and aerospace engineering and have been used for decades (Simpson, et al. [11]). It is well
known that design problems in engineering fields are much more well-defined than those encountered
in architectural design. For example, the overall appearance of an aircraft has not changed significantly
over the past decades because the overall design is heavily governed by physical laws and principles.
Nowadays, the fundamental variables describing these laws and principles are well understood by
practitioners in the field and thus, result in more robust implementation of techniques such as
metamodels. The same cannot be said for the field of architectural design however, because the design
space of a building is vague and wide open, and can involve the exploration of many variables and
different problem descriptions. In this context, we imply that when importing techniques such as
metamodels, from engineering fields to architectural design, we need to accommodate for flexibility, to
address the iterative nature of creativity in design.
We hypothesise that through collaboration between designers and engineers, we can achieve a
synthesised metamodel that addresses the engineering aspects controlling the physical behaviour
directly, without diving too deep into the laws of physics. In turn, the metamodel becomes more
fundamental to the problem domain thus generalises for different design problems within that domain.
In this context, we propose to shift from adopting design variables as the metamodel inputs, towards
selecting variables that are known to drive the simulation response and are thus design problem-
independent. As a result, the design-independent metamodel becomes more general and is thus, able to
accumulate data and information from one problem to the next (Figure 2). Furthermore, the
generalizability of the inputs can also be taken advantage of to build a metamodel from multiple data
sources (Figure 2), for example, from past and present design projects.
In theory, the proposed approach applies to any type of metamodel however, in this paper we retain
focus on building a flexible knowledge-oriented metamodel using a Bayesian Network. Furthermore,
we will focus on the design of 2D truss structures using FEA as the structural analysis of choice. Figure
2 illustrates the overall workflow in building a flexible metamodel.
In the following section we will discuss an example to illustrate building a flexible metamodel using our
approach. We will focus on explaining steps 1, 3A and 4A from Figure 2.
x2=?
x1=?
xk=?
y=0
…
x2
x1
xk
y
…
x2=4
x1=2
xk=1
y=?
…
bayesian network
metamodel (BNM)
specify
hard input
values
predict
response PD
infer input
PD
B
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
PDs of
sampled
values
PDs of
simulation
responses
y = f(X)
typical
metamodel
inference with BNM
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
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0
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0
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4
0
1
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3
4
0
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3
4
specify hard
response value
B1
B2
A

Proceedings of the IASS Symposium 2018
Creativity in Structural Design
4
Figure 2: Workflow to build a flexible BNM, that can generalise for new design inputs.
3.1. Case study problem: 2D truss design and analysis using FEA
To demonstrate our approach, we assume three different parametric 2D cantilever truss designs, each
defined by a respective set of design variables (as indicated in Figure 3). A parametric analysis model
of each structure was modelled in Grasshopper and then analysed using ‘Millipede’ (Michalatos and
Kaijima [12]), which is a Grasshopper plugin for FEA-based structural analysis. For all problems we
assume solid round steel cross-sections with a density of 7800 *)+,-, Young’s Modulus of 200 ./0 ,
fixed conditions in ', # and rotational directions on the left side, and a vertical point load of 10012
acting in the -# direction, at the far right node/s.
Figure 3: (a) symmetric, (b) asymmetric and varying cross-sections, (c) michell truss.
3.2. Identifying and extracting the metamodel input variables (Steps1, 3A)
The scope is to select variables that are (i) known to be critical in the calculation of the simulation
response, or known to drive the physical behaviour, and are (ii) general, i.e. independent of the design
STEP 3A
STEP 1
PARAMETRIC
FEM
EXTRACT SIMULATION
RESPONSE VALUES
y
STEP 4A
STEP 3B
STEP 4B
BUILD
METAMODEL
STEP 2
EXTRACT VALUES FROM
IMPORTANT VARIABLES
XE1, … , XEn
XE1
XE2
XEn
XE3
…
xn
x1
x2
STEP 5
SELECT IMPORTANT
VARIABLES
DESIGN VARIABLES
EXPLORE NEW
DESIGN
VARIABLE/S ?
YES
A B
METAMODEL
INPUTS
METAMODEL
OUTPUT
xk
xi
xii
xz
xa
xb
DESIGN VARIABLES
PARAMETRIC
FEM
RUN PARAMETRIC FEA
y
INPUT SAMPLE MATRIX SINPUT SAMPLE MATRIX S
y
x
SPAN [2m , 6.5m]
DESIGN PROBLEM A
100KN
SPAN [2m , 6.5m]
MIN_RADIUS
[40mm , 100mm]
f ( THETA )
[0.5 , 2]
DESIGN PROBLEM B
100KN
NUM SEGMENTS
[5 , 15]
SPAN
[4m, 10m]
TENSILE_RAD
[50mm , 200mm]
COMPRESSIVE_RAD
[50mm , 200mm]
BOUNDARY_RAD
[50mm , 200mm]
DESIGN PROBLEM C
100KN
DEPTH
[0.5m, 2m]
NUM
SEGMENTS
[2 , 6]

Proceedings of the IASS Symposium 2018
Creativity in Structural Design
5
problem. In this step, we encourage a collaboration with domain experts as to help with identifying
suitable metamodel input variables. The following is merely an example for this paper, to illustrate the
type of variables that could be used for generalising a metamodel thus, we emphasise that our
suggestions are subject to improvement based on further expert consultation. For our example, we take
a peek at what constitutes the Finite Element (FE) method, which is the numerical calculation underlying
FEA-based structural analysis.
3.2.1. Finite element method background
The description of the laws of physics driving engineering phenomena are usually expressed in terms of
partial differential equations (PDEs). However, when complicated geometries, loadings and material
properties are involved, it becomes impossible to solve these PDEs analytically. Instead, methods such
as the FE method are adopted to construct approximations of the solutions of the PDEs, by discretising
the problem domain and solution into a set of smaller parts, called ‘finite elements’, that can be solved
numerically as a set of algebraic equations. Together, the finite elements make up a finite element model.
In general, the algebraic equation of a linear finite element model is expressed as 3 4 $ 5 , where
4 is the unknown vector of nodal deformations, 3 the stiffness matrix and 5 the vector of external
forces, that depends on the loading conditions. These vectors and forces are assembled from the
respective contributions of each element 6 given by the element nodal deformation vector 7489, element
stiffness matrix 38 and element load vector 7589. The stiffness matrix of each finite element contains
the material and geometric information that indicates the resistance of the element to deformation when
subjected to loading. In the case of 2D beam elements, the deformation may be constituted of axial and
bending effects. The local axial stiffness of an element depends on the Young’s Modulus of the material
:, the cross-sectional area ;, and length of the element <, and can be expressed in terms of :;+<, while
the local bending stiffness depends on :, moment of inertia of the element = and <, and can be expressed
as :=+<. Furthermore, the local deformation of each element is transformed to the global axis in which
the load is acting, such that global axial stiffness becomes%>:; ?@ ABC D in '-direction, and
>:; ?@ CEF D in #-direction, while global bending stiffness becomes >:= ?@ ABC D in '-direction, and
>:= ?@ CEF D in #-direction, where D is the angle between the global x-axis and the neutral axis of the
element. Subsequently, the transformed stiffness matrices from each element are assembled together
into the global stiffness matrix that takes into account the connectivity between the elements, and the
node deformations are calculated by solving the global system 4 $ 3 GH 5.
3.2.3. Identify important variables (step 1)
For this example, we can therefore identify >:; ?@ ABC D, >:; ?@ CEF D, >:= ?@ ABC D,
and%>:= ?@ CEF D as suitable variables for our flexible metamodel because they are (i) critical to the
calculation of the simulation response, and (ii) are independent of the design variables. Our scope is not
to assemble the global stiffness matrix, but to select metamodel inputs. Therefore, we treat the values of
each important variable as an accumulation of all the elements. Furthermore, we decide to also include
additional variables; the total mass of the assembled structure and the center of gravity in ' and #
directions (cogx and cogy, respectively). Even though they are already implicitly considered in the
bending and axial stiffness calculations, we argue that additional independent metamodel inputs might
benefit the model to capture information that is being ‘compressed’ when accumulating stiffness values.
3.2.4. Extract the important variables from FEM (step 3A)
In our example, we obtain ;, ?, D%and = for each element, directly from the parametric model where, D
is calculated as the dot product between the neutral axis of the element and the global '-axis, and = is
obtained using the parallel axis theorem to find the moment of inertia of each element with respect to
the center of gravity of the global structure. Finally, the Young’s Modulus :, is prescribed and kept
constant in our example.

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Creativity in Structural Design
6
3.3. Building the generalised base BNM
In this section, we first build the base metamodel from data generated with problem A. Subsequently, in
section 4, we carry forward the same metamodel to problems B and C (Figure 4).
The simulation response data for the base metamodel is generated by sampling and evaluating the design
space of problem A, using a quasi-random sampling algorithm as per typical metamodeling practice.
The difference with our approach is that on each simulation run, besides the maximum deflection, we
also store values for >:; ?@ ABC D, >:; ?@ CEF D, >:= ?@ ABC D, >:= ?@ CEF D, total mass, cogx and
cogy. Each time, new design variables are explored, the new generated data is concatenated with the
previous dataset and the metamodel is then rebuilt, at no significant computational expense.
Subsequently, all past and new continuous data is discretised (as required by Bayesian Networks), and
introduced as probability distributions (nodes) in a Bayesian Network. The edge direction of causal
structure is interpreted from input to output nodes (Figure 4). Subsequently, the probabilistic
relationships between the nodes (marginal and conditional probabilities) are then learned automatically
from the discretised data by means of a supervised learning EM algorithm. For this research we made
use of ‘libpgm’ (CyberPoint International [13]), which is a Python library for modelling Bayesian
networks and performing inference. Once the base metamodel is built, design variables are then mapped
onto the metamodel inputs in the form of a secondary Bayesian Network such that the metamodel can
then be used to predict response, and/or infer the design input distributions for a target response values
of interest. The subsequent design problems are mapped onto the base model in the same way.
Figure 4: Workflow to map new design inputs onto the generalsied BNM.
3.3.1. Cross-validate the base BNM
In order to test the robustness of the metamodel for predicting simulation response, we perform a ten-
fold cross validation, which splits the dataset into training/testing, in ten different ways. Since the output
of the BNM is non-scalar, we cannot make use of scalar prediction error measures such as mean square
error. Instead we plot the distribution of differences between the mean of the predicted distribution bin
and the actual simulated value. Figure 5 illustrates three of the ten folds and demonstrates that for a
metamodel based on 2000 samples, the BNM predicted responses within ~10% of their actual value.
Figure 5: Prediction robustness of base BNM for 2000 samples from design problem A.
PROBLEM A
GENERALISED METAMODEL
INPUT DESIGN SPACE A
EA/L
sinθ
EA/L
cosθ
EI/L
cosθ
µ
max
span
num
segs
EI/L
sinθ
mass
cogx
cogy
EA/L
sinθ
EA/L
cosθ
EI/L
cosθ
EI/L
sinθ
mass
cogx
cogy
[ENGINEERING DOMAIN]
[DESIGN DOMAIN]
INPUT DESIGN SPACE B
INPUT DESIGN SPACE C
DESIGN INPUT VARIABLE
FUNDAMENTAL VARIABLE
FORWARD INFERENCE
TO PREDICT RESPONSE
REVERSE INFERENCE
TO PREDICT
DESIGN INPUTS
LEGEND
RESPONSE VARIABLE
t_
rad
EA/L
sinθ
EA/L
cosθ
EI/L
cosθ
EI/L
sinθ
mass
cogx
cogy
c_
rad
b_
rad
num
segs
PROBLEM C
PROBLEM B
theta
span
min_
rad
EA/L
sinθ
EA/L
cosθ
EI/L
cosθ
EI/L
sinθ
mass
cogx
cogy
depth
span
Frequency
% Prediction Error % Prediction Error % Prediction Error
Fold_0 Fold_6 Fold_8
Frequency
Frequency

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Creativity in Structural Design
7
4. Using the generalized base BNM to explore new design spaces
In order to demonstrate the generalizability of the base BNM built in the previous section, we will carry
forward and build on the same model to predict response and infer inputs, for two new truss design
problems: B and C (Figure 3). For each problem we illustrate plots to demonstrate (a) robustness for
predicting maximum deflection and (b) an example of reverse inference to predict the input distributions
for minimised max deflection. See Xuereb Conti and Kaijima [1], for interpreting the inferred PDs.
4.1. Input design problem B (3 design variables - 1000 new simulation runs)
For this problem, we generate 1000 simulation runs from problem B, and concatenate them to the
previous dataset, to produce a dataset of 3000 data points. Subsequently, we rebuild the BNM from the
concatenated dataset of 3000 data points. As a validation technique, we use the rebuilt BNM to predict
the maximum deflection for each of 2000 combinations of span, theta and min_radius values, which the
model has not seen before. Figure 6, left demonstrates that the generalised BNM predicts the maximum
deflection values within more or less 10% from the correct simulated values. Figure 6, right illustrates
the use of the BNM to explore the input distributions that would likely yield a max_deflection < ~1mm.
Figure 6: Prediction validation (left), example of inferring input distributions (right).
4.1. Input design problem C (5 design variables - 500 new simulation runs)
In this example, we carry forward the metamodel from problem B to a problem with five design
variables. We concatenate 500 new simulation runs from problem C, to the previous 3000. We show
that despite the increased number of design variables, the histogram (Figure 7, left) indicates that the
model has the potential to predict within decent error range, considering the addition of only 500 new
data points from problem C. Figure 7, right illustrates the use of the BNM to explore input configurations
that yield a max_deflection < ~4mm.
It is important to note that the number of new data points, is highly dependant on the the quality of the
selected metamodel inputs.
Figure 7: Prediction validation (left), example of inferring input distributions (right).
Frequency
max_defletion span theta min_rad
% Prediction Error
P(def) <
0.927mm
marginal PD
inferred PD
marginal PD
inferred PD
marginal PD
inferred PD
marginal PD
Probability
Probability
Probability
Ranges (mm) Ranges (m) Ranges Ranges (m)
Probability
Frequency
% Prediction Error
Frequency
Probability
Probability
Ranges (mm)
Probability
Ranges (m)
Probability
Ranges (m)
Probability
Ranges
Probability
Ranges (m) Ranges (m)
boundary_radspan
max_defletion
compression_radtensile_raddensity
P(def) <
3.93mm
marginal PD
inferred PD
marginal PD
inferred PD
marginal PD
inferred PD
marginal PD
inferred PD
marginal PD
inferred PD
marginal PD
Probability

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Creativity in Structural Design
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5. Conclusion and future work
In this paper we presented an approach to build a metamodel that can generalise for new problems such
that it can either be used to carry data forward from one design space exploration to the next, or be used
to pipe multiple sources of existing data into one metamodel. We achieve this flexibility through a
careful selection of the metamodel inputs. Our approach requires a collaboration with domain experts in
search for important variables that are (i) critical for the calculation of the simulation response, and (ii)
independent of the design variables. As a result, in our case study we demonstrated that we can
accumulate data from one problem to then next with the same metamodel, which in turn has the potential
to reduce the amount of simulation data for subsequent design problems. The latter depends on the
quality of the selected metamodel inputs however, the sub-hypothesis requires further investigation.
In future work, we would like to push the generalizability of the metamodel further such that it can
generalise for different boundary, loading, and material scenarios. Furthermore, we would like to focus
on smarter sampling strategies, such that we can predict more specifically where in the design space the
samples should be mostly concentrated to avoid redundant sampling that is already captured form data
of previous problems.
Acknowledgements
We would like to thank Dr. Oliver Weeger and Dr. Narasimha Boddeti from the Digital Manufacturing
and Design centre at SUTD, for their expert insight in aspects of structural mechanics.
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