This 3 sentence summary provides the high level information from the document: The document discusses integration schemes for fast hybrid testing used at the University of Colorado's NEES facility. It describes how an unconditionally stable implicit integration method, specifically the α method, is used to achieve real-time or close to real-time earthquake simulations through a constrained implementation. Key aspects of the integration scheme include maintaining displacement continuity and force equilibrium between the numerical and experimental components throughout the numerical integration process.

1. CU-NEES-06-8
NEES at CU Boulder
The George E Brown, Jr. Network for Earthquake Engineering Simulation
01000110 01001000 01010100
The CU-Boulder Fast Hybrid Test
Integration Schemes for
Fast Hybrid Testing
by
Dr. Eric Stauffer
Technical Director
Department of Civil Environmental and Architectural Engineering
September 2006 University of Colorado
UCB 428
Boulder, Colorado 80309-0428

2. 1 ABSTRACT 1
1 Abstract
The Fast Hybrid Testing (FHT) system at the University of Colorado (CU) enhances the con-
ventional pseudodynamic testing method by facilitating real-time or close to real-time with
consistent scaling of the temporal component of earthquake simulations. The CU FHT system
achieves a rate of loading that is significantly higher than that of conventional pseudodynamic
testing and has achieved hard realtime for a variety of test configurations. In general hybrid
simulation presents a broad set of challenges in that both a numerical and experimental struc-
ture are ultimately simultaneously involved and indeed interacting in the simulation. Until
fairly recently realtime hybrid simulation was not feasible owing to computational and tech-
nological limitations. Advances in realtime networking technology, realtime operating systems
and computation efficiency have made realtime earthquake simulation within the context of a
hybrid model possible. This paper summarizes recent advances in the techniques utilized at the
CU NEES Fast Hybrid Testing Facility in particular the numerical integration scheme which
is so central to FHT. The FHT system at CU is part of the George E. Brown, Jr. Network for
Earthquake Engineering Simulations (NEES).
2 Introduction
Central to the task of hybrid simulation is the numerical engine that drives the experimental
component in unison with the numerical component of a test. The modular nature of finite
element modeling is ideally suited to this task and has been used widely for hybrid testing.
In the past these techniques where called pseudo-dynamic and involved an every changing
distortion of the time scaling of a simulation. This distortion is the result of the variability
in the time it takes to complete the computation for each time step and then update the
command signals controlling a portion of the displacement field for the experimental component.
This approach remains the state of the art in many research laboratories throughout the US.
However, with the increased availability of greater computational power, realtime networking,
and realtime operating systems it is possible to maintain a consistent scaling of time throughout
a simulation. Limited by computational speed and model complexity hard realtime is in fact
possible. Realtime here implies a 1-to-1 scaling of prototype vibration to simulation vibration
with no temporal distortion.
Direct integration methods are employed to establish equilibrium and displacement con-
tinuity at discrete intervals of time specified by a time step . A wide variety of integration
techniques are standard features in most FEM software packages. Fundamentally these tech-
niques can be categorized as either explicit or implicit each having inherent limitations and
capabilities. The explicit schemes are attractive in there relative simplicity and numerical effi-
ciency but place a limitation on the size of the time step owing to issues of numerical stability.
The implicit schemes are typically more complex, involving iteration for the nonlinear case,
but having more favorable stability characteristics. In fact for linear models and with properly
selected integration parameters it has been established that implicit schemes such as the α
method are unconditionally stable and despite the presence of high frequencies within a model
an arbitrarily large time step may be used. For this reason the method has been selected as
the basis for the direct integration routine used in the CU NEES FHT system.
In applying direct integration techniques to a consistently time scaled hybrid simulation it
is necessary to place constraints and special features in the numerical implementation. Most of
these constraints and features are a result of particular needs stemming from the experimental
CU-NEES-06-08 CU-Boulder FHT Integration Schemes

3. 3 BACKGROUND 2
or physical component of a hybrid simulation. The need for these constraints will be explained
in more detail shortly. These constraints and features require access and modification to the in-
tegration algorithm and other aspects of the FEM source code. Currently the CU NEES facility
is using OpenSees for most of its hybrid simulation needs. OpenSees is a fully object oriented
modeling framework intended primarily for the earthquake engineering research community
and is open-source. OpenSees is supported by a small staff of programmers at the University
of California Berkeley and remains affiliated with the Pacific Earthquake Engineering Center
(PEER). Recently at the CU NEES facility effort has been directed toward the generalization
of the added hybrid code segments such that they can be used within other FEM codes or act as
a basis for the development of a new code that is specifically designed for the unique demands
of hybrid simulation.
3 Background
The pusedodynamic test method originally developed by Takanashi et al (1975) [?] provides
a systematic means of recreating earthquake like loading on a test component by directly
integrating a discrete representation of the governing equations of motion. The reduction of the
structure from a continuum to a finite set of discrete equations may be achieved by application
of the finite element method resulting in a second order ordinary differential equation.
Ma + Cv + r(x) = f (1)
where M and C are the mass and viscous damping matrices for the idealization of the
structure, r the restoring force vector resulting from deformation x, and f is the vector of ap-
plied forces due to a seismic event or some other dynamic stimulus. Typical application of the
pseudodynamic test method proceeds on the basis that M and C are known and understood
sufficiently to be represented purely numerically while some portion or perhaps the entirety of
the restoring force is determined experimentally due some uncertainty or potentially complex
nonlinear behavior. For the conventional pseudodynamic test method the physical test compo-
nent is subjected to displacements on relevant degrees of freedom in some quasi-static manner,
typically ramp and hold. The duration of the ramp and hold phase of each time step may be
set arbitrarily large to accommodate inadequacies in the performance of the testing equipment,
communication latency, and/or computational speed. Recently efforts have focused on faster
and perhaps continuous application of motion to the physical test component (Magonnete 2001,
Nakashima et al. 1992, Horiuchi et al. 1996, Darby et al. 1999, Shing et al. 2002, Mosqueda et
al 2005). These efforts culminate with simulations that are conducted in realtime and preserve
the original rate of loading thereby reducing the compromising effects of load relaxation, re-
duced and/or inconsistent strain rates and relaxed or neglected similitude laws. Rate sensitive
devices such as semi-active dampers, i.e. magnetorheological dampers, require that hybrid sim-
ulations be conducted in realtime. The Fast Hybrid Testing (FHT) system at the University of
Colorado NEES facility utilizes a customized unconditionally stable implicit integration scheme
(Hilber, Hughes and Taylor 19??) to achieve fast and continuous motion. Realtime performance
has been achieved for several different experimental test configurations, most recently involving
200 K Newton MR dampers. The method used to maintain displacement continuity and force
equilibrium throughout the numerical integration process for the hybrid test structure will be
the focus of this paper.
CU-NEES-06-08 CU-Boulder FHT Integration Schemes

4. 4 A FAST HYBRID TESTING SYSTEM WITH REALTIME CAPABILITIES3
4 A Fast Hybrid Testing System with Realtime Capabilities
The hybrid simulation capabilities at the CU NEES facility are based on a constrained imple-
mentation of the α method (Hughes, 1983). The favorable stability and damping properties
of this method and its successful application to conventional pseudodynamic tests (Shing et
al. 1991) make it well suited for a fast and continuous testing system. With the understand-
ing that the externally applied force vector is balanced by inertial, damping, and restorative
force components and that each of these components is composed of contributions from both
the numerical (FEM) and experimental portions of the hybrid simulation the damping term is
generalized so as to admit nonlinear behavior. In so doing equation 1 is modified to include the
nonlinear damping term s
Ma + s(v) + r(x) = f. (2)
When considering a hybrid structure each of the three terms Ma, s(v) and r(x) may be
expanded so as to make explicit the hybrid nature of this representation.
Ma = (Mexp + MFEM )a (3)
s(v) = sexp(v) + CFEM (v) (4)
r(x) = rexp(x) + rFEM (x) (5)
The discrete time equilibrium equations for this representation of a second order dynamic
system are
Mai+1 + (1 + α)si+1 − αsi + (1 + α)ri+1 − αri = (1 + α)fi+1 − αfi (6)
di+1 = di + ∆tvi + ∆t2 1
2
− β ai + βai+1 (7)
vi+1 = vi + ∆t [(1 − γ) ai + γai+1] (8)
Where M is the mass matrix, s is the damping force vector assumed to be a nonlinear
function of the velocity vector, r and f are the restoring force and external applied force
vectors respectively.
This direct method of time integration determines equilibrium at equally spaced time inter-
vals which herein will be referred to as the integration interval. In order to allow for nonlinear
structural response it necessary to include an iteration capability that converges to the equilib-
rium condition within each integration interval. A modified Newton-Raphson iteration method
is applied to the discrete equations of motion (equation 2). The finite number of iterations,
which will be constrained to a fixed and constant number (Shing et. Al 2002) act to subdivide
the integration interval into n iteration intervals. If each interval is given equal time weighting
the iteration interval may be expressed as
δt = ∆t/l (9)
Where l is the integer number of Newton iteration intervals, and are the time intervals
associated with the integration interval and iteration intervals respectively. By fixing l to be
a constant integer value a favorable degree of determinism is achieved which is important for
realtime integration and hybrid testing. This determinism comes at the price of constraining
CU-NEES-06-08 CU-Boulder FHT Integration Schemes

5. 4 A FAST HYBRID TESTING SYSTEM WITH REALTIME CAPABILITIES4
1nx
nx
3
1nx
2
1nx
1
1nx
t Prototype Time (seconds)
Simulation Time (seconds)
1nt nt 1nt
t
t
x
Figure 1: Command Interpolation during Newton Iteration
the calculation of equilibrium to a limited number of Newton iterations and a fixed interval of
time in the case of consistently scaled and realtime simulations. Experience at the CU NEES
FHT facility has indicated that ∆t = 0.01 and δt = 0.001 are reasonable values that balance
the need for accuracy and speed.
A simplified discrete representation of the force equilibrium equation in residual form is
obtained by solving 7 for ai+1 and substituting into equation 6
fr = Mdn+1 + ce + c1(sn+1 + rn+1) (10)
where
ce = −M[dn + ∆tvn + ∆t2
(0.5 − β)an] − ∆t2
β[α(sn + rn − fn) + (1 + α)fn+1] (11)
c1 = ∆t2
β(1 + α) (12)
Equation 10 has two unknowns, sn+1 and vn+1 , which are independent variables for the
damping and stiffness terms respectively. Each is treated as general nonlinear relationship. By
combining equations 7 and 8 an equation expressing vn+1 in terms of known quantities and
dn+1 is obtained
vn+1 =
γ
β
1
∆t
(dn+1 − dn) + (
β
γ
− 1)vn + ∆t
β
γ
−
1
2
an (13)
With equations 10 and 13 a general modified Newton iteration procedure can be used to
solve for the unknown discrete displacement field dn+1. The iterative solution procedure is
based on the linearized Taylor series representation of the residual equilibrium equation
fr(dn+1 + ∆d) ≈ fr(dn+1) +
∂fr
∂dn+1
∆d. (14)
Where successive displacements increments ∆d are computed until a convergence criterion
is satisfied. With appropriate consideration of the nonlinear damping and stiffness terms the
Jacobian may be expressed in the form
∂fr
∂dn+1
= M + c1
∂sn+1
∂vn+1
∂vn+1
∂dn+1
+
∂rn+1
∂fn+1
(15)
CU-NEES-06-08 CU-Boulder FHT Integration Schemes

6. 5 CONCLUSION 5
For the purpose of efficiency within the numerical integration both the stiffness and the
damping terms are approximated with the linearized initial stiffness and initial damping ma-
trices Ki and Ci.
∂fr
∂dn+1
= M + c1 Ci
∂vn+1
∂dn+1
+ Ki. (16)
Finally a single equation is obtained which is repeatedly solved until the desired level of
accuracy is obtained
dk+1
n+1 = dk
n+1 − m + c1(
α
∆tjβ
Ci + Ki)
−1
fr(dk
n+1). (17)
The indices n and k indicate the integer value for the time step and the Newton iteration
number respectively. In preliminary testing this new integration scheme has proven to be equal
or superior to the prior scheme (restricted to linear viscous damping) both in terms of accuracy
and rate of convergence.
5 Conclusion
The George E. Brown Jr. Network for Earthquake Engineering Simulation (NEES) is made up
of 15 advanced research laboratories at universities in the US and is intended to advance the
state of the art in earthquake engineering research. These 15 laboratories are linked to each
other with grid software that facilitates collaboration, remote participation, distributed testing,
and a single data archive that acts as a long term repository for test results and documentation.
In 2001 the NSF selected the University of Colorado as one of the 15 prominent universities to
receive support under this program. The facility at CU specializes in realtime or Fast Hybrid
Testing (FHT) and its application to vibration testing and simulation in earthquake engineering.
The FHT Facility at the University of Colorado has, by design, been developed with real-
time hybrid simulation capabilities in mind. Realtime hybrid simulation which synchronously
combines numerical and experimental test components into a hybrid test setting must operate
within an efficient and deterministic computational environment. To achieve this all critical
computational, control and measurement systems utilize realtime operating systems and are
networked to one another with a realtime shared memory network. Additionally, all physical
testing hardware which is currently exclusively servo-hydraulic must also have high-performance
dynamic capabilities. The need for high performance capabilities must not compromise the pre-
cise and stable control of the multi-actuator testing hardware. This is achieved using state of
the art hydraulic equipment provided by MTS Corporation. This custom equipment and testing
technology is maintained and operated by a skilled professional staff that was also integral to
the development and commissioning of this unique testing system. The direct time integration
scheme presented here is specifically designed for realtime hybrid simulations.
6 Acknowledgements
The financial support of CU-NEES is gratefully acknowledged.
CU-NEES-06-08 CU-Boulder FHT Integration Schemes