proper orthogonal decomposition framework for the explicit ... › ~anil-lab › others › lectures...

Chiara CeccatoDepartment of Civil, Architectural

and Environmental Engineering,

University of Padua,

Padua 35131, Italy

Xinwei ZhouEngineering and Software System

Solutions, Inc. (ES3),

San Diego, CA 92101

Daniele PelessoneEngineering and Software System

Solutions, Inc. (ES3),

San Diego, CA 92101

Gianluca Cusatis1Department of Civil and Environmental

Engineering,

Northwestern University,

Evanston, IL 60208

e-mail: [email protected]

Proper OrthogonalDecomposition Framework forthe Explicit Solution of DiscreteSystems With SofteningResponseThe application of explicit dynamics to simulate quasi-static events often becomesimpractical in terms of computational cost. Different solutions have been investigated inthe literature to decrease the simulation time and a family of interesting, increasinglyadopted approaches are the ones based on the proper orthogonal decomposition (POD)as a model reduction technique. In this study, the algorithmic framework for the integra-tion of the equation of motions through POD is proposed for discrete linear and nonlin-ear systems: a low dimensional approximation of the full order system is generated bythe so-called proper orthogonal modes (POMs), computed with snapshots from the fullorder simulation. Aiming to a predictive tool, the POMs are updated in itinere alternat-ing the integration in the complete system, for the snapshots collection, with the integra-tion in the reduced system. The paper discusses details of the transition between the twosystems and issues related to the application of essential and natural boundary conditions(BCs). Results show that, for one-dimensional (1D) cases, just few modes are capable ofexcellent approximation of the solution, even in the case of stress–strain softening behav-ior, allowing to conveniently increase the critical time-step of the simulation without sig-nificant loss in accuracy. For more general three-dimensional (3D) situations, the paperdiscusses the application of the developed algorithm to a discrete model called latticediscrete particle model (LDPM) formulated to simulate quasi-brittle materials character-ized by a softening response. Efficiency and accuracy of the reduced order LDPMresponse are discussed with reference to both tensile and compressive loading conditions.[DOI: 10.1115/1.4038967]

1 Introduction

One of the main issues in computational mechanics is related tothe capability of modeling the behavior of highly nonlinear struc-tures subjected to dynamic and quasi-static loading when the inte-gration of the governing equations leads to very expensivenumerical simulations. If a large number of degrees-of-freedom(DOF) and nonlinearities are involved, explicit time integrationtechniques are generally preferred to implicit methods and consid-ered attractive not only for wave propagation problems, wherethey mostly found applications, but also for slow dynamics andquasi-static problems [1,2]. The main benefit of explicit algo-rithms is that they are not affected by convergence problemswhich may occur when dealing with nonsmooth phenomena, likecontact, softening damage laws or cohesive zone models, anddamage localization. Furthermore, they usually require a lightercomputational effort per each time-step, since full mass and stiff-ness matrices are not assembled. Explicit algorithms, however,are not unconditionally stable and require an accurate evaluationof the numerical stability. In particular, since the time-stepdecreases with the highest natural frequency of the computationalsystem, a prohibitive number of time increments are required forproblems governed by low frequencies. Indeed, in many cases,time steps much larger than the critical one are required for a cer-tain accuracy but cannot be used due to stability requirements

(see, among others, [1,3–5]). Mass scaling [6], for instance, wasdeveloped to increase the time-step size by adjusting the mass ofthe most critical elements; significant errors, though, can originateif those elements where the mass scaling is applied have a signifi-cant contribution to the global system response and, consequently,more elaborated techniques need to be applied [7,8]. Dynamiccondensation has also been used in the literature for this purpose(see e.g., Ref. [9]). Another interesting and promising approach isthe application of the proper orthogonal decomposition (POD) asa model reduction technique, which has now found applications indifferent fields of engineering and physics (e.g., Refs. [10–18])following the pioneeristic studies of Refs. [19–21]. Dynamic sys-tems can be projected onto subspaces containing the solution ofthe problem or a good approximation of it, so that a high-dimensional process is converted into a low-dimensional one [22].

The POD application to complex nonlinear problems has beenpursued also by some researchers and different numerical techni-ques are now being investigated to improve both efficiency andaccuracy of POD approaches [23–26].

The aim of the present study is to investigate the use of theproper orthogonal decomposition as model reduction techniquefor the explicit integration of low-dynamic or quasi-static prob-lems featuring strain softening behavior.

2 Explicit Integration of Equations of Motion

In this section, the algorithmic framework for the solution ofthe dynamics equation through a reduced order method is dis-cussed. Discrete Systems are of particular concern in this study,being focused on the applicability of model reduction techniques

1Corresponding author.Contributed by the Applied Mechanics Division of ASME for publication in the

JOURNAL OF APPLIED MECHANICS. Manuscript received November 27, 2017; finalmanuscript received January 5, 2018; published online March 7, 2018. Editor:Yonggang Huang.

Journal of Applied Mechanics MAY 2018, Vol. 85 / 051004-1Copyright VC 2018 by ASME

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to the lattice discrete particle model (LDPM), a discrete modeldeveloped to simulate quasi-brittle materials (see Refs. [27–33])and briefly reviewed in Sec. 5.1. More generally, though, the pre-sented algorithm can be applied to any discrete dynamic equilib-rium problem solved through explicit integrators such as discreteelements methods ([34–36]) and discrete molecular dynamics(DMD) [37,38]. Let us consider the equations of motion in Rn fora discrete (or discretized) system

M€u þ f intðuÞ ¼ fext (1)

defined 8t � [t0, tf] and with initial conditions uðt0Þ ¼ u?;_uðt0Þ ¼ _u?. In Eq. (1), M is the mass matrix, u is the DOF vector(containing nodal displacements and, if relevant, rotations), fint isthe internal forces vector, fext is the external forces vector, and u

?

and _u? denote the initial displacements/rotations and correspondingvelocities, respectively. Dirichlet boundary conditions (BCs) are

applied on a portion, B, of the boundary in the form _uiðtÞ ¼_�uðtÞ 8xi 2 B � Rn.

The system 1 needs to be discretized in time to be solvednumerically. For this purpose, the middle point rule is quite often

used and it gives €um ¼M�1½fmext � f intðumÞ�; _umþ1=2 ¼ _um�1=2þ€umDt, and umþ1 ¼ um þ _umþ1=2Dt with Dt being the time stepand m the number of the current time-step, with m 2 ½1; nsteps�;nsteps 2N�. In the elastic regime, stability requirements limit thetime-step Dt to be subjected to the constraint Dt

and Eq. (1) can be projected onto Rk through Bk. One obtainsMBk €d þGBk _d þ f intðBkdÞ ¼ fext.

The matrix defines the so-called POMs which can be inter-preted as shape function with global support. By premultiplying

the left-hand and right-hand sides of this equation by BTk , thefollowing system of equations is obtained:

BTk MBk€d þ BTk f intðBkdÞ ¼ BTk fext (6)

where BTk f int and BTk MBk are the projections of the internal force

vector and mass matrix, respectively, in S ¼ spanðBkÞ � Rk.The reduced order system of equations can be integrated

numerically with the middle point rule in the reduced subspaceRk through the following algorithm:

€dm ¼ ðBTk MBkÞ

�1ðfmext � fmintÞ (7)

and _dmþ1=2 ¼ _dm�1=2 þ €dmDtR; _d

mþ1=2 ¼ _dm�1=2 þ €dmDtR Theunknowns are the coefficients di (i¼ 1, k), which define the ampli-tude of each orthogonal deformation mode used to approximatethe actual solution. In each time-step, u and _u are evaluated fromthe associated projected values and the internal forces, and theresidual forces are computed in the full order. The time-step ofthe reduced system DtR is subjected to a different constraint com-pared to Dt: DtR< 2/xk,max, where xk,max is computed from thefollowing eigenvalue problem: detðBTk KBk � x2kBTk MBkÞ ¼ 0.

3.4 Essential Boundary Conditions. The POMs satisfy auto-matically any homogeneous essential (Dirichlet) BCs by construc-tion. In fact, the U matrix has a zero-valued row corresponding toany fixed degree-of-freedom and this information is transferred tothe modes themselves and, consequently, to any object in the sub-space they span.

Nonhomogeneous essential BCs must be re-applied in thereduced system, since this information is not automatically pre-served by POMs. It is not easy, however, to apply the BCs directlyto the projected degrees-of-freedom in the reduced subspace,because this leads to an overdetermined system. To overcome thisproblem, the nonhomogeneous essential BCs can be applied indi-rectly as equivalent external forces through the penalty method:fext¼ kp(up� u), where fext is the external force equivalent to theBC, kp is the penalty coefficient, and up is the penalty displace-ment prescribed by the BC. The penalty enforcement of theboundary conditions in the reduced order model has been previ-ously explored by Kalashnikova and Barone [44].

Due to instability issues, the node mass M where the penaltyconstraint is applied must be artificially increased (mass scaling)by adding a quantity which is proportional to the penalty coeffi-cient kp and to the square time-step Dt: M

scal¼Mþ 1.1kpDt2.When the penalty method is applied, the computation of the stabletime-step in the projected system must also take into account theincreased stiffness due to the penalty constraint.

3.5 Mode Updating. The snapshots can be computed a poste-riori, after the end of the regular analysis. This method allows thedefinition of the reduced system for model validation purposesand the snapshot collection can be the most efficient possible,since the real solution is already known. Aiming to a predictivetool, though, the snapshots should be collected during the analysisitself, in itinere, alternating the integration in the complete system,for the snapshots collection, with the integration in the reducedsystem, until the snapshots previously collected cease to be agood representation of the actual response. When dealing withquasi-static problems, the complete system can be integrated onlyfor a small initial time interval DT, just to capture an adequatenumber of snapshots, which sufficiently describe the systembehavior in average. The corresponding reduced system can, then,be computed from the first k POMs and the analysis carried out in

the reduced system with a larger time-step. Obviously, the snap-shots may need to be updated to take into account any variation ofthe external input (for instance, in case of changes in appliedforces, displacements, velocities) or nonlinearities (for instance,due to nonlinear constitutive law).

After a time interval DTR, when the POMs are not representa-tive anymore, the analysis could be carried out back through thecomplete system to recompute a new set of POMs. Differentamplitudes of DT and DTR lead to different accuracy levels of theapproximation, since the definition of the reduced order subspaceand the frequency of its updates are directly affected. Once thesnapshots have been collected and the corresponding reducedspace defined, the integration in the reduced system can start,

using as initial condition the following values, _dm�1 ¼ BTk _um�1

and dm�1 ¼ BTk um�1, where _u and u are the values of the nodalunknowns computed in the last time-step of the complete systemintegration.

As already explained, the stable time-step in the reduced systemis higher than the one in the complete system DtR>Dt. To allow aproper transition from the integration scheme in the completespace to the reduced one and vice versa, an appropriate time-stepdefinition must be used. The time-step DtV¼ (DtRþDt)/2 is usedfor the integration of velocities in both first and last time-step inthe reduced system. For the displacement integration, instead,DtD¼DtR is used for the first time-step and DtD¼Dt is used forthe last time-step. Figures 1(b) and 1(c) clarify the time steps defi-nition in the transition from one integration scheme to the other,considering that the velocities are defined at half interval accord-ing to the middle point rule, whereas the displacements aredefined at the end of the interval.

Also, when the integration shifts from the reduced system backto the complete one, in order to compute new POMs, the initialcondition for velocity and displacement to be used in the full orderspace integration is computed as

_um ¼ 1DtD

Bk dm � dm�1ð Þ (8)

Equation (8) provides the initial conditions for the integration inthe complete system. While the displacement is computed directlyfrom the projection matrix B, the velocity is averaged from thedisplacements in the relevant interval. This approach was shownto provide a smooth transition of the system response from thereduced to the complete integration, as opposed to the option of

using as initial condition the following expression: _um ¼ Bk _dm

.In the latter case, abnormal oscillations were observed in the veryinitial part of the response after the transition to the completesystem.

4 One-Dimensional Implementation and Analysis

4.1 System Description. In order to explore the potentialapplicability of the POD to nonlinear problems and to understandhow this technique might be used to extract qualitative andquantitative information about the response of large mechanicalsystems, a simple one-dimensional (1D) benchmark example isdescribed here. As shown in Fig. 2, a simple discrete system isconsidered in which a number n of masses mi¼ qAil are connectedto each other (and to a wall at one end) by nonlinear springs.

In the elastic regime, the spring force is proportional to therelative displacements of the two adjacent nodes: Pi¼ kidi,di¼ uiþ1� ui where ki¼EAi/li, E is the elastic modulus, and Aiand li are area and length, respectively, of the ith spring. The load-ing condition consists of an imposed constant velocity, v0, appliedto the last node on the right-hand side of the system.

During the softening regime, instead, the spring force Pi is com-

puted incrementally as _Pi ¼ ki _di; _di ¼ _uiþ1 � _ui and is subjectedto the following constraints �1 � Pi � Pbi ðdÞ, where

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Pbi ¼ Airt expð�Hthdi � rt=kii=rtÞ; Ht ¼ 2E=ðlt=li � 1Þ, and lt isthe material characteristic length.

For the benchmark study discussed later, the following parame-ters are selected: E¼ 30,000 MPa, rt¼ 3 MPa, lt¼ 100 mm, L ¼P

i li ¼ 10 cm with the total number of springs n being variable,Ai¼A¼ 5 cm2, q¼ 2500 kg/m3, and v0¼ 10 mm/s.

4.2 Linear Elastic Behavior. The main goal of this section isto understand how the accuracy of the approximation and the min-imum time-step required for the stability are affected by thedimension of the full order system. Systems with 26, 51, 101, and201 DOFs (corresponding to number of springs¼ 25, 50, 100,200) are considered in the analysis and they are solved for a totaltime ttot¼ 0.01 s.

The number of snapshots to be collected for the accurate defini-tion of the POMs depends on the problem being solved. As far asthe elastic behavior is concerned, a small number of snapshotshomogeneously distributed in the interval of interest is enough forthe purpose. The results in this section are obtained with 20equally spaced snapshots collected in the first 2� 10�4 s corre-sponding to 2% of the total simulation time.

Figure 3 shows the shape of the first 6 POMs. The shape of themodes is independent of the number of degrees-of-freedom sinceeven the system with 26 DOFs has enough resolution for thesemodes. Of course, this is not necessarily the case for highermodes. The POMs can be interpreted as shape functions withglobal support: if the eigenvectors are properly normalized, thenthe coefficients of the linear combination used for the integrationof the reduced system can be considered the displacement frac-tions associated with that mode.

The first POM is a straight line, which can be considered as thestatic response of the system, while the other modes account forthe vibrations due to dynamical effects.

The ratio between the stable time-step in the reduced system,DtR, and the stable time-step in the original system, Dt, increaseswith the number of DOFs and, consequently, the so-called

Fig. 1 (a) Time step scheme and (b) and (c) definition of time steps in the transition from complete to reduced integration (b)and from reduced to complete integration (c)

Fig. 2 One-dimensional model of a dynamic discrete system

Fig. 3 Shape of the first six POD modes

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performance improvement factor (PIF), also increases with thenumber of DOFs [3]. PIF is defined as the ratio between the totalCPU time, T, needed to solve the complete system and the totalCPU time, Tr, required to solve the reduced system, including thesnapshots collection and the singular value decomposition compu-tation: PIF¼T/Tr

It is worth pointing out that the stable time-step during thereduced order integration is a decreasing function of the penaltystiffness. Hence, larger values of the penalty stiffness, while pro-viding more accurate results in terms of the application of theBCs, lead to smaller values of the PIF. For the simulations dis-cussed in this section, the penalty stiffness was set to kp¼ 103km,where km ¼ maxiðkiÞ. The accuracy of the reduced order modelcan be assessed by calculating the external energy error, e, at theend of the analysis. That is e¼ 100(W*�W)/W in which W andW* are the external work in the complete and reduced systems,respectively, at the end of the simulation.

If only the first POM is used for the reduced system, the follow-ing results are found: DtR/Dt¼ 4.5, 12.5, 34.0, and 90.0;PIF¼ 6.7, 19.4, 46.1, and 112.2; e¼ 0.8%, 0.6%, 1.2%, and 2.0%;for 26, 51, 101, and 201 degrees-of-freedom, respectively.

The analysis of the results shows that the first mode is sufficientto represent accurately the response of the system in terms of dis-placements, forces, and energy. It accounts for more than 99.9%of the total sum of all eigenvalues, independent of the number ofdegrees-of-freedom of the original problem. The gain in terms ofthe PIF increases when the number of DOFs increases. For eachnode, during the whole time history, the displacements are wellapproximated, especially with reference to the average (quasi-static) response by the POD algorithm.

Figure 4 compares, for the 51 DOFs system, the full andreduced order solutions in terms of load (calculated at the fixedend) versus applied displacement curves for 1 POM, 3 POMs, and5 POMs. As shown in Fig. 4(a), the first mode describes the sys-tem behavior in average, filtering out all the high frequency vibra-tions, which do not provide significant enhancement in accuracyespecially if one is interested in capturing the quasi-staticresponse. It is important to mention that this result is not generalbut rather problem dependent. Other problems with differentmaterial behavior and loading conditions might require moremodes as it will be evident later in this paper.

Obviously, the higher the number of modes considered byPOD, the greater the subspace where the integration takes placeand, thereby, the closer the projected subspace to the full spacecontaining the real solution. So, as more modes are included inthe subspace definition, the POD algorithm is capable of repre-senting a higher number of details in terms of system response.

Figures 4(b) and 4(c) illustrate this behavior. By adding twomore modes, the solution is able to represent some vibrations in

the response and the following modes allow the approximate solu-tion to include more vibrations. When the number of modes isequal to the number of degrees-of-freedom of the problem, thePOD solution and the real solution coincide. However, as thenumber of modes increases, the stable time-step of the analysisbecomes smaller because the POD subspace size becomes closerto the size of the full space: DtR¼ 12.5 Dt for 1 mode, DtR¼ 6.0Dt for 3 modes, and DtR¼ 4.0 Dt for 5 modes. By enlarging thesubspace, the stable time-step DtR reduces progressively and itbecomes exactly equal to Dt when all POMs are used.

4.3 Softening Behavior. The same geometry, materialparameters, and BCs described in Sec. 4.1 are considered in thissection. The simulations are relevant, unless otherwise specified,to the system with 51 DOFs and with POMs computed, as for thelinear elastic case, with 20 snapshots equally spaced over a fullorder simulation interval of 2� 10�4 s. In order for the fractureprocess to be realistically activated, one spring (the central one) isassigned a reduced strength of rt¼ 2.5 MPa. The overall responsefeatures a softening postpeak branch and deformation localizationin the weaker spring.

As long as the tensile stress in each element is smaller than thetensile strength, the behavior of the system is linear elastic andwhat was observed in Sec. 4.2, for the elastic behavior, identicallyholds. However, when the tensile strength of the weaker elementis reached, the softening mechanisms are activated and the frac-ture process starts. The elastic subspace, originated from the snap-shots collected at the beginning of the analysis, cannot representwell the real solution anymore and a new adequate subspace needsto be defined for the POD algorithm to continue. The spectralmodes computed before fracture occurs do not allow for the dis-placement discontinuity due to fracture to be represented. As aconsequence, the overall response of the reduced order system issignificantly more ductile than that of the full order system ascomparison of the relevant load versus displacement curves dem-onstrates in Fig. 5(a).

Therefore, integration of the full order system must resume justbefore the maximum tensile stress is reached in the weak springand a new set of snapshots in the softening branch, just before andafter the point of maximum tensile stress, must be collected forthe computation of a new proper orthogonal subspace. While themodes computed in the elastic regime do not account for the dis-placement discontinuity due to fracture, the new modes do asshown by the first two modes plotted in Figs. 6(a) and 6(d).

The first mode alone, however, is still not sufficient because asthe displacement at the right end of the system increases the dis-placement discontinuity (crack opening) at the fracture locationand the displacement gradient away from the discontinuity bothincrease (see Figs. 6(b) and 6(c)). Since away from the

Fig. 4 Force versus displacement curves for the linear material and zoom in for (a) 1 mode, (b) 3 modes, and (c) 5 modes(51 DOFs)



discontinuity the material behavior is elastic, the displacementgradient is proportional to the stress which, consequently,increases in the softening regime in contrast with the stressdecrease required by the softening law in the fracturing spring andthe series coupling of all the springs in the system. The obtainedload versus displacement response is again more ductile than thefull order one (Fig. 5(b)) and the force distribution along the sys-tem becomes more and more imbalanced as the simulationprogresses.

A different behavior is observed when the reduced integrationuses two POMs. For this case, Figs. 6(e) and 6(f) report the dis-placement distribution along the system coordinate at two selectedtimes at the beginning of the softening regime and during the soft-ening regime. As the softening progresses, the displacement dis-continuity increases while the slope of the displacement outsidethe discontinuity decreases. With 2 POMs, the reduced orderresponse approximates very well the full order solution also interms of load versus displacements curve as shown in Fig. 5(c).

Figure 5(d) shows that using the same subspace with 2 POMs,unloading–reloading behavior can be also reproduced with noneed to switch to the complete system in order to update thePOMs. It is worth noting that the unloading–reloading rule used inthe calculations and featuring a large hysteresis is not supposed tobe representative of any real material behavior but just an exampleto demonstrate how loading–reloading conditions can be handledby the proposed POD approach.

The aforementioned strategy cannot be applied if the time atwhich the mode updating is required is a priori unknown or cannotbe estimated. In this case, it is possible to automatically updatethe snapshots switching to the complete system after a predeter-mined number of time intervals. More frequent snapshots updat-ing leads to more accurate reduced order solution but to lessreduction of the computational cost. Setting the time interval for

the fully explicit algorithm to run and the snapshots to be col-lected equal to DT¼ 0.25 ms and the time interval for the reducedintegration with 2 POMs equal to DTR¼ 0.55 ms (12 updates),e< 3% can be achieved in the softening branch (Fig. 5(e)) withPIF¼ 3.2. If the number of updates is reduced, the accuracy alsoreduces, as shown in Fig. 5(f) in which the snapshots are updated8 times during the reduced integration, using DT¼ 0.25 ms andDTR¼ 0.95 ms, with an energy error of around 18%. When thereduced order solution deviates significantly from the full ordersolution, at the transition between the two integration schemes,the equilibrium needs to be reestablished dynamically and thisleads to an excessive oscillatory behavior.

As described earlier, the reduced order simulations of the non-linear behavior provide an accurate approximation of the responsewith just 2 POMs. It is important to point out, however, that thenumber of snapshots collected for the definition of the subspace ofinterest plays also an important role in terms of accuracy of thesolution with reference to the local values of forces, especially ifthe POMs are not updated periodically. To investigate this aspect,Fig. 7 reports the results of simulations based on POMs computedby collecting 20, 50, and 300 snapshots out of the 600 availablefrom the full order simulation. The snapshots are chosen to behomogeneously spaced in the interval of the update in order tocapture the trend of the response in that selected interval. As onecan see, if the snapshots collected are not representative enoughof the overall behavior, the solution is sought in a subspace whichis too limited and, therefore, the system is forced with additionalconstrains which are generated from the restricted dimension ofthe subspace where the solution can be sought, and consequently,it needs to be equilibrated by increased internal forces.

This is the reason for the wide variation of the internal forcealong the system coordinate reported in Fig. 7(a) for the case of20 snapshots and two simulations with 2 and 20 POMs. As one

Fig. 5 Force versus displacement curves for the softening material (a) without modes update, (b) with modes update and 1mode, (c) and (d) with modes update and 2 modes, (e) with 12 automatic updates, and (f) with 8 automatic updates (51 DOFs)



can see, such variation depends very little on the number ofPOMs. However, the response converges to the full-order solutionby increasing the number of snapshots as illustrated in Fig. 7(b)for 50 snapshots and 7(c) for 300 snapshots. Analysis of theresults in terms of error en provides similar conclusions. For 2POMS, en¼ 0.2%, 0.8%, and 6% for 300, 50, and 20 snapshots,respectively; for 20 POMS, en< 0.1%, 0.3%, and 4.5% for 300,50, and 20 snapshots, respectively. The number of snapshots hasnegligible influence on the PIF, which is mainly affected by thedimension of the subspace, given by the number of POMs.

Finally, it is interesting to investigate the improvement in com-putational cost when the size of the full-order system increases.

For comparison, the simulations are performed by updating thePOMs around the peak. For systems with 16, 51, 101, and 201DOFs (corresponding to number of springs n¼ 25, 50, 100, and200), one has DtR/Dt¼ 4, 9, 25, and 60; PIF¼ 4.8, 13.2, 20.2, and45.5; e¼ 0.5%, 0.2%, 0.6%, and 0.4%, respectively. Similar to theelastic case, the improvement of the computational cost increaseswith the size of the system making the proposed reduced orderapproach best suited for large computational systems.

4.4 Mass Scaling and Proper Orthogonal Decomposition.A further improvement of the reduced system efficiency in termsof computational gain can be obtained by the use of mass scaling

Fig. 6 (a) Shape of the first POM and displacements distribution along the system (b) after 1.0 ms (d 5 0.01 mm applied dis-placement) and (c) after 10 ms (d 5 0.1 mm applied displacement) using only the first POM in the reduced integration, (d) shapeof the second POM and displacements distribution along the 1D system (b) after 1.0 ms (d 5 0.011 mm applied displacement),and (e) after 10 ms (d 5 0.1 mm applied displacement) using the first and second POMs in the reduced integration

Fig. 7 Force distribution trend along the 1D system at the beginning of softening (d 5 0.025 mm applied displacement) for dif-ferent numbers of POMs and snapshots (out of 600)



combined with POD. Applying the mass scaling during thereduced integration only, the snapshot computation and the POMsremain unchanged, while the critical time-step DtR in the corre-sponding subspace is allowed to increase arbitrarily large. Obvi-ously, accuracy considerations are required in order to obtainsatisfactory results.

The scaled mass matrix for the reduced integration is computed

as aBTk MBk, where a ¼ Dt2Rx2max=4 and x2max is the highest eigen-value of the matrix ðBTk MBkÞ

�1BTk KBk. Combining POD andmass scaling provides a considerable computational saving and itallows an optimal compromise between accuracy and increasedtime-step. For an elastic material, Fig. 8 highlights the improvedefficiency of the mass scaling technique when coupled with theproper orthogonal decomposition. The mass scaling techniquealone is not accurate enough (see Figs. 8(a) and 8(b)) when theincreased time-step DtR is much bigger than the actual stabletime-step Dt, because the increased mass causes the developmentof high inertial forces. When the mass scaling is combined withPOD, this effect is much smaller. Figures 8(c) and 8(d) show thecomparison of the full order response in terms of global load ver-sus displacement curve and displacement versus time curve for anode in the middle of the system for DtR¼ 50Dt. Similarly, Figs.8(e) and 8(f) report the same comparison for DtR¼ 1000Dt. Asone can see, for both cases, the POD mass-scaled response is basi-cally overlapped with the full order response. Minor deviationsappear only for very large time steps as shown in Fig. 8(f). It isworth observing that even in the case of mass scaling, the internalenergy is various order of magnitude larger than the kineticenergy. This ensures that the explicit dynamic response is a goodapproximation of the quasi-static response.

The POD mass-scaling strategy discussed earlier can also beused in conjunction with the modes update approach needed for

softening response. Figures 9(a)–9(c) report the relevant compari-son with the full order simulation and for DtR/Dt¼ 10, 25, and 50.

5 Three-Dimensional Application to the LatticeDiscrete Particle Model

In this section, the POD framework is applied to three-dimensional(3D) simulations performed with the LDPM in order to evaluatethe potential of the reduced-order algorithm for more complexnumerical models when fracture and other nonlinear phenomenaare involved.

5.1 Brief Review of Lattice Discrete Particle Model.LDPM was proposed for the first time by Cusatis and coworkersin 2011 [27,45] to simulate the behavior of quasi-brittle materialsat the mesoscale level by modeling the interaction of material het-erogeneities. The geometrical mesostructure of the material isobtained through the following steps: (1) Material heterogeneitiesare assumed to be spherical particles and are introduced into thespecimen volume V by sampling an assumed particle size distribu-tion function, which, for example for cementitious composites, isderived from a set of mix-design parameters (cement content c,water-to-cement ratio w/c, aggregate-to-cement ratio a/c, maxi-mum aggregate size da, minimum aggregate size d0, governing themodel resolution, and Fuller coefficient nf). (2) Zero-radius par-ticles are randomly distributed over the external surfaces to facili-tate the application of boundary conditions. (3) Delaunaytetrahedralization of the generated particle centers and the associ-ated three-dimensional domain tessellation are then carried out toobtain a network of triangular facets inside each tetrahedral ele-ment as shown in Fig. 10(b). A portion of the tetrahedral elementassociated with one of its four nodes I and the corresponding

Fig. 8 Mass scaling and POD for elastic response: (a) DtR 5 50 Dt (mass scaling only), (b) DtR 5 1000 dt (mass scaling only), (c)and (d) DtR 5 50 dt, and (e) and (f) DtR 5 1000 Dt



facets are shown in Fig. 10(c). Combining such portions from alltetrahedral elements connected to the same node I, a correspond-ing polyhedral particle is obtained. Each couple of adjacentpolyhedral particles interacts through shared triangular facets(Fig. 10(a)). The triangular facets, where strain and stress quanti-ties are defined in vectorial form, are assumed to be the potentialmaterial failure locations. Three sets of equations are necessary tocomplete the discrete model framework: definition of strain oneach facet, constitutive equation which relates facet stress vectorto facet strain vector, and particle equilibrium equations.

Facet strain definition. Rigid body kinematics is employed todescribe the deformation of the lattice/particle system. The dis-placement jump, ½½uC��, at the centroid of each facet is used todefine measures of strain as

ea ¼1

r½½uC�� ea (9)

where a¼N, M, L (eN is the facet normal strain component; eMand eL the facet tangential strain components); r is the length ofthe line that connects the nodes sharing the facet and correspond-ing to the associated tetrahedron edge (see Fig. 10(b)); ea are unitvectors defining a facet local Cartesian system of reference suchthat eN is orthogonal to the facet, and eM and eL are the facet tan-gential unit vectors (see Fig. 10(c)). It was recently demonstrated[32,46] that this definition of strain is consistent with the projec-tion of the classical micropolar strain tensor into the local systemof reference attached to each facet.

Facet vectorial constitutive equations. A vectorial constitutivelaw governing the behavior of the material is imposed at the

centroid of each facet. Formally, one can write t¼ f(eN, eM, eL)where t ¼ tNeN þ tMeM þ tLeL is the traction vector applied oneach triangular facet. Details of the constitutive equations used inthis work are reported in Appendix.

Particle equilibrium equations. Finally, the governing equa-tions of the LDPM framework are completed through the equilib-rium equations of each individual particle. Linear and angularmomentum balance equations for a generic polyhedral particlecan be written as

XF

Atþ Vb0 ¼ 0;XF

Aw ¼ 0 (10)

where F is the set of facets that form the polyhedral particle, A isthe facet area, V is the particle volume, b0 is the body force vector,and w is the moment of t with respect to the node which is locatedinside the polyhedral particle.

Lattice discrete particle model is implemented in a computa-tional software named MARS [47] and was used successfully tosimulate concrete behavior in different types of laboratory experi-ments [45]. Furthermore, LDPM has shown superior capabilitiesin modeling concrete behavior under dynamic loading [28,48],alkali silica reaction deterioration [49–51], fracture simulation offiber reinforced polymers (FRP) reinforced concrete [52], failureof fiber-reinforced concrete [53–55], and early age behavior ofultra high performance concrete [30,56,57]. LDPM was alsorecently formulated to simulate sandstone [58], shale [31,59], andwaterless concrete [29].

5.2 Direct Tension Test. This section presents the applica-tion of the POD approach to the analysis of a direct tension test.The simulations were performed on a dogbone shaped specimen(Fig. 11(a)) of 100 mm height, 30 mm thickness, 100 mm (widerends) and 50 mm (narrower cross section) widths. A constantvelocity of _d ¼ 10 mm=s was applied to the nodes belonging tothe top surface by using the penalty approach described earlier inthis paper. Figure 11(b) shows the fracture pattern obtained in atypical LDPM simulation.

The simulations used initially a coarse LDPM system (304nodes, 1824 DOFs) in order to evaluate the effect of the numberof snapshots used to create the reduced order space and the num-ber of POMs used in the reduced order calculations during the ini-tial elastic response and the subsequent fracture process. Differentnumbers of snapshots (out of the 1000 available in the chosenintervals) were collected at the beginning of the simulation (fromd¼ 0 mm to d¼ 0.004 mm) and then updated at the end of thelinear elastic phase (from d¼ 0.01 mm to d¼ 0.02 mm, see Fig.12(a)). As one can see in Figs. 12(a) and 12(b), a large number ofsnapshots and a large number of POMs are required in order to

Fig. 9 Load versus displacement curves for mass scaling and POD: (a) DtR 5 10 Dt, (b) DtR 5 25 Dt, and (c) DtR 5 50 Dt (51 DOFs,softening material)

Fig. 10 (a) LDPM polyhedral particle enclosing sphericalaggregate pieces; (b) typical LDPM tetrahedron connecting fouradjacent aggregates and its associated tessellation; and (c) tet-rahedron portion associated with aggregate I



capture the softening behavior in the LDPM simulations. Whilethe linear branch is easily captured with a limited number of snap-shots and of POMs, this is no longer true when the fracturing pro-cess starts and the POMs are updated only once. The progressivedevelopment of cracks requires a larger subspace in order to becorrectly represented and a large number of snapshot for its com-putation. The accuracy of the solution from the reduced systemimproves progressively by enlarging the subspace computed fromthe same set of snapshots (Fig. 12(a)) and, likewise, by increasingthe number of snapshots used for a subspace of the same dimen-sion (Fig. 12(b)).

It is possible to keep the subspace small while achieving a rea-sonable accuracy if the POMs are updated frequently enough toprevent the loss of accuracy and to capture the softening branch.Indeed, the problem is that during the fracture evolution, a set ofcalculated modes soon become no longer representative of theongoing response due to the initiation and coalescence of manycracks in 3D. Therefore, the POMs need to be recomputed from anew set of snapshots in the full order system. Figure 12(c) showsthe results in case the POMs are automatically updated everyDd¼ 0.005 mm increment of displacement. With this approach,the error with respect to the reference (full order) solution is keptacceptably small. In this case, one obtains DtR/Dt¼ 3.5, PIF¼ 1.7,and e¼ 9%. The computational efficiency is relatively modestbecause of the small size of the system. For a larger system with505 nodes (3003 DOFs), the automatic update of the POMs results

in DtR/Dt¼ 5.0, PIF¼ 2.9, and en¼ 11%. The application of theproposed approach to very large systems (with hundreds of thou-sands of DOFs) is expected to lead to much larger improvementof the computational efficiency.

5.3 Compression Tests. An even more challenging problem,addressed in this section, is relevant to compression tests. In thiscase, especially in the absence of transverse confinement, thecrack pattern is quite complex and evolves significantly in thenonlinear regime. Simulations were performed on unconfined andconfined cylinders of 100 mm length and 50 mm diameter. Thesamples consisted of 90 nodes (540 DOFs) and were loaded witha compressive velocity _d ¼ 10 mm=s.

Figures 13(a) and 13(b) compare, for the unconfined case, thePOD solutions computed in subspaces of increasing dimensions.The snapshots were collected at the beginning of the simulationsand updated at the peak load (from d¼ 0.2 mm to d¼ 0.3 mm). Inthis case, 500 snapshots (out of the 2500 available) were collectedin the interval of interest.

One can see in Fig. 13(a) that the number of POMs necessaryfor an accurate approximation is considerably high (450 POMsout of 540 DOF for the considered example) and the accuracydecreases as the number of spanning modes decreases. From amechanical point of view, the reason for this reduced efficiencycan be related to the different failure modes of the material in

Fig. 11 Dogbone shaped specimen for direct tension test: (a) LDPM geometry and (b)fracture pattern

Fig. 12 Load versus displacement curves from the fully explicit simulation and the POD algorithm (a) with different numbers ofPOMs and fixed number of snapshots (1000), (b) with different numbers of snapshots and fixed number of POMs (900), and (c)with automatic updates (2 POM and 10 snapshots)



tension and compression. As a matter of fact, when a concretedogbone specimen is loaded in pure tension up to failure, damagelocalization occurs. On the contrary, in the case of a concrete cyl-inder loaded in unconfined compression, the fracture process leadsto a complex three-dimensional crack system. It is worth observ-ing that when a limited number of POMs are used, the response ischaracterized by a more gradual softening during the postpeak.This is the result of a “confinement effect” due to the fact that thesolution is sought in a reduced order subspace which cannot con-tain the full order solution so that the algorithm provides addi-tional artificial constraints to the system.

Figure 13(b) shows the effect of the number of snapshots on theaccuracy of the reduced order response for simulations performedwith 100 POMs. One can see that, in this case, the accuracy of thereduced order approximation depends mostly on the number ofPOMs rather than the number of snapshots.

Similar to the case of tension, better results can be obtained byan automatic mode updating strategy. Figure 13(b) shows thePOD response obtained with a 2 POMs built from 10 snapshotshomogeneously distributed in the collection interval (Dd¼ 0.034 mm, with 850 snapshots available) and automaticallyupdated every Dd¼ 0.1 mm of vertical displacement. The PODresponse is characterized by DtR/Dt¼ 3.0, PIF¼ 1.7, and en¼ 6%.

The simulation of confined compression consisted of the sameconcrete cylinders simulated for unconfined compression wrappedwith FRP sheets, which were simulated according to Ref. [52].Figure 13(d) confirms, once again, that the number of POMsrequired for an accurate simulation of the inelastic behavior isconsiderably high, almost coincident with the degrees-of-freedomof the system. The accuracy improves also by increasing the num-ber of snapshots collected in the update interval, but with aweaker effect (Fig. 13(e)). One can observe that the slope of theinelastic branch of the load versus displacement curve increases

progressively with the decreasing dimension of the subspace gen-erated by POMs due to the numerical “confinement effect” pro-vided by the limited number of modes spanning the subspace.Finally, Fig. 13(e) shows the response with automatic update andcharacterized by DtR/Dt¼ 85, PIF¼ 6.8, and en¼ 3%.

It is worth mentioning that the POD approximation is moreaccurate for the FRP confined-compression test than for theunconfined compression test and a higher stable time-step can beachieved in the former case. Indeed, in the inelastic phase fracturelocalization, events are less pronounced when the specimen is lat-erally confined and the crack pattern is more homogeneous. Forthis reason, the reduced order solution tends to be more accurate.

5.4 Conclusions. This paper discussed the reduced ordersolution of discrete systems with softening response by means ofPOD. Of particular interest was the application of the POD to thesimulation the quasi-static behavior of such systems with anexplicit dynamic algorithm. Based on the presented results, thefollowing conclusions can be drawn:

(1) The proper orthogonal decomposition is a powerful tool tobuild reduced order approximations of the response of largesystems, both linear and nonlinear, solved with explicitdynamics algorithms.

(2) Characteristic spectral modes are computed by collectingsnapshots of the full order response in certain timeintervals.

(3) The spectral modes serve the role of shape functions withglobal support that approximate the actual system deforma-tion with a reduced number of degrees of freedom.

(4) The spectral modes constrain the high-frequency deforma-tion modes of the full order system leading to a larger sta-ble time step for the explicit integration of the reduced

Fig. 13 Load versus displacement curves for the concrete cylinder under unconfined and confined compression, from the fullyexplicit simulation and the POD algorithm: (a) unconfined compression simulations with increasing number of POMs (500 snap-shots), (b) with increasing number of snapshots (100 POMs), (c) with automatic updates (2 POMs, 10 snapshots), (d) confinedcompression simulations with increasing number of POMs (500 snapshots), (e) with increasing number of snapshots (100POMs), and (f) with automatic updates (2 POMs, 10 snapshots)



order equations of motion. However, full advantage of theincreased time step is observed mostly in the case whenonly few spectral modes are used.

(5) Accuracy and efficiency of the reduced order model dependon the number of snapshots used to build the reduced orderspace and on the number of spectral modes used in the sim-ulations. However, a large number of used snapshotsincrease the computational cost to build the reduced orderspace and such increase quickly offsets the computationalgain of the reduced order integration.

(6) Homogeneous essential boundary conditions are automati-cally transferred from the full order system to the reducedorder system. Nonhomogeneous essential boundary condi-tions can be imposed through a penalty algorithm.

(7) Significant reduction in computational cost can be obtainedby combining POD with classical mass-scaling approaches.

(8) The spectral modes must be updated when nonlinear behav-ior leads to a significant change on the deformation charac-teristics of the system. This is particularly important for 3Dapplications with softening that are characterized by com-plex crack patterns. The best results are obtained by a peri-odic update of the spectral modes during the simulations.

(9) Better results are expected for the 3D case if mass scalingis combined with POD. This topic is being investigated bythe authors in the ongoing studies.

Funding Data

� The National Science Foundation (Grant No. CMMI-1435923).

� ES3 R&D resources.

Appendix: Lattice Discrete Particle Model ConstitutiveEquations

Full details of the LDPM constitutive equations used in thispaper are reported in Refs. [27,45,52]. In the elastic regime, thenormal and shear stresses are proportional to the correspondingstrains: tN ¼ ENeN ; tM ¼ ETeM; and tL ¼ ETeL, where EN¼E0 andET¼ aE0 with E0 being the effective normal modulus and a theshear–normal coupling parameter. For stresses and strains beyondthe elastic limit, concrete mesoscale nonlinear phenomena arecharacterized by three mechanisms and the corresponding facetlevel vectorial constitutive equations are here briefly described.

Fracture and Friction Due to Tension and Tension–Shear.For tensile loading (eN> 0), the fracturing behavior is formulatedthrough an effective strain, e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2N þ aðe2M þ e2LÞ

p, and stress,

t ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2N þ ðt2M þ t2LÞ=a

p, which are used to define the facet normal

and shear stresses as tN¼ eN(t/e); tM¼ aeM(t/e); and tL¼ aeL(t/e).The effective stress t is incrementally elastic ( _t ¼ E0 _e) and mustsatisfy the inequality 0 � t � rbt(e, x), where

rbt ¼ r0ðxÞexp ½�H0ðxÞhemax � e0ðxÞi=r0ðxÞ� (A1)

in which hxi ¼ maxfx; 0g; e0ðxÞ ¼ r0ðxÞ=E0; tanðxÞ ¼ eN=ffiffiffiap

eT ¼ tNffiffiffiap

=tT in which eT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2M þ e2L

pand tT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffit2M þ t2L

p. x

is the parameter that defines the degree of interaction between

shear and normal loading. emax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2N;max þ ae2T;max

qis a history

dependent variable, and emax¼ e in the absence of unloading. Thepost-peak softening modulus is defined as H0ðxÞ ¼ Htð2x=pÞnt ,where nt is the softening exponent, Ht is the softening modulus inpure tension (x¼ p/2) expressed as Ht¼ 2E0/(lt/r� 1);lt ¼ 2E0Gt=r2t ; and Gt is the mesoscale fracture energy. LDPMprovides a smooth transition between pure tension and pure shear(x¼ 0), with a parabolic variation for strength

r0ðxÞ ¼ rtr2stð�sinðxÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2ðxÞ þ 4a cos2ðxÞ=r2st

qÞ

� 1= ½2a cos 2ðxÞ� (A2)

Compaction and Pore Collapse in Compression. Normalstresses for compressive loading (eN< 0) are computed throughthe inequality �rbc(eD, eV) � tN � 0, where rbc is a strain-dependent boundary that depends on the volumetric strain, eV, andthe facet deviatoric strain, eD¼ eN� eV. The volumetric strain iscomputed by the volume variation of the tetrahedral element aseV¼DV/3V0 and is assumed to be constant for all facets belongingto a given tetrahedron. Beyond the elastic limit, rbc models porecollapse for (ec0 �� eV � ec1) as a linear evolution of stress forincreasing volumetric strain with stiffness Hc and compaction andrehardening beyond the pore collapse limit for (�eV ec1).ec0¼ rc0/E0 is the compaction strain at which pore collapse starts,and ec1¼ jc0ec0 is the compaction strain at the beginning of rehar-dening. jc0 is a material parameter and rc0 is the mesoscale com-pressive yield stress. Therefore, one can write

rbc ¼rc0 for � eV < 0rc0 þ h�eV � ec0iHc for 0 � �eV � ec1rc1 exp ½ð�eV � ec1ÞHc=rc1� otherwise

8<: (A3)

where Hc ¼ ðHc0 � Hc1Þ=ð1þ jc2hrDV � jc1iÞ þ Hc1; rDV ¼jeDj=eV0 for (eV> 0); and rDV ¼ �jeDj=ðeV � eV0Þ for (eV � 0), inwhich eV0 ¼ jc3ec0; rc1ðrDVÞ ¼ rc0 þ ðec1 � ec0ÞHcðrDVÞ, andrc0; Hc0; Hc1; jc1; jc2; jc3 are material parameters.

Friction Due to Compression-Shear. The incremental shearstresses in the presence of compression are computed as _tM ¼ETð_eM � _epMÞ and _tL ¼ ETð_eL � _e

pLÞ, where _e

pM ¼ _k@u=@tM; _e

pL ¼

_k@u= @tL, and k is the plastic multiplier with loading–unloadingconditions u _k � 0 and _k 0. The plastic potential is defined asu ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffit2M þ t2L

p� rbsðtNÞ, where the nonlinear frictional law for

the shear strength is assumed to be

rbs ¼ rs þ ðl0 � l1ÞrN0½1� expðtN=rN0Þ� � l1tN (A4)

where rN0 is the transitional normal stress; l0 and l1 are the ini-tial and final internal friction coefficients, respectively.

In the simulations presented in the paper, the following LDPMmesoscale parameters were used: c¼ 280 kg/m3 (cement content),w/c¼ 0.77 (water to cement ratio), a/c¼ 7.5 (aggregate to cementratio), nf¼ 0.5 (Fuller curve exponent); E0¼ 40,000 MPa (normalelastic modulus), a¼ 0.25 (shear–normal coupling parameter),rt¼ 3.65 MPa (tensile strength), lt¼ 200 mm (characteristiclength), rs/rt¼ 2.5 (shear-to-tensile strength ratio), nt¼ 0.2 (soft-ening exponent), rc0¼ 45 MPa (yielding compressive stress), Hc0/E0¼ 0.3 (initial hardening modulus in compression), jc0¼ 4(transitional train ratio), jc1¼ 1 (first deviatoric-to-volumetricstrain ratio), jc2¼ 5 (second deviatoric-to-volumetric strain ratio),l0¼ 0.2 (internal friction coefficient), l1¼ 0 (internal asymptoticfriction), rN0¼ 600 MPa (transitional stress), and Ed/E0¼ 1 (den-sified normal modulus).

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