AIAA-92-2327-CP

AN ADAPTIVE MULTI-LEVEL SUBSTRUCTURING METHOD FOR EFFICIENT MODELING OF COMPLEX STRUCTURES

Jeffrey K. Bennighof and Connie K. ~ i m ~

The University of Texas a t Austin Austin, Texas 78712

A b s t r a c t

An adaptive method for modeling complex struc- tures with maximum accuracy with a given number of degrees of freedom is presented. T h e method uses a multi-level substructuring approach, in which the structure is partitioned into a number of substructures, each of which is itself composed of substructures, and so on. This approach lends itself to efficient represen- tation of localized response in particular, and also to parallelization. Convergerrce using this method is sig- nifc;tnt,ly faster t,hari when the model is rcfiried u ~ i i - forrrily, i n terms of the number of degrees of freedom required to achieve a given level of accuracy. A nu- merical example is presented.

I n t r o d u c t i o n

Many aerospace and naval structures possess such complex geometry that any models tha t represent them with a reasonable level of detail must have an extremely high number of degrees of freedom. Finite element models with tens or hundreds of thousands of degrees of freedom are used increasingly frequently for static structural analysis, but for harmonic response over a range in frequency, such models can require computation that is prohibitively expensive. In the structural acoustic analysis of naval structures, low- frequency modeling can be done using a fairly crude model of an entire structure, because elastic wave- lengths are long enough tha t fine details are incon- sequential. High-frequency modeling can be done suc- cessfiilly by modeling the region around the excitation in detail, but treating portions further from the ex- citation as if they were rigid, because of the localiza- tion of the response. IIowever, i n the middle frequency range, response can be sigriificant over the entire struc- ture, and can involve short enough elastic wavelengths tha t the entire structure must be modeled with a pro- hibitively high level of detail.

For problems such as these, extending the fre- quency range that can be handled will require methods that can achieve increased accuracy with a given num- ber of degrees of freedom. Because of the camplexity of the behavior tha t is of int.erest, i t does not seem feasible to reliably predict a priori how model order

'Assistant Professor, Aerospace Engineering & Engineering hlechanics. Member A IAA.

tGraduate Research Assistant.

can be reduced to a minimum without dcgrading accu- racy. Guyan reduction' and substructuring methods2 are used frequently in dynamic structural analysis, but primarily a t very low frequencies where the analyst can comfortably rely on intuition.

For higher frequencies, it quickly becomes attrac- tive t o shift the burden of model reduction to the com- puter as much as possible, so an automated adaptive method is of interest. Adaptive methods have been used in more and more areas in which finite elements are used over the past decade,3'" but they have found limited application in structural analysis. Existing adaptive methods refine the finite element model by subdividing elements or by increasing the degree of the polynomial interpolation in elements so tha t the solu- tion of partial differential equations is approximated more accurately. However, in structural analysis, a crude model must be refined so tha t it represents the geometric detail more accurately. Even though the response is governed by partial differential equations over members of the structure, for practical purposes the response can be considered t o be governed by an extremely high-order discrete model which represents all of the geometric detail of the struct,ure very accu- rately. If further refinement beyond this high-order detailed model is necessary, existing adaptive meth- ods could be appropriately used. However, for many problems refinement of the model to represent the gec- metric detail of the structure requires so many degrees of freedom that existing adaptive methods would never be used.

The above considerations suggest a new approach for adaptive analysis of complex structures. In this approach, the highest level of det,ail t ha t is likely to be needed is used t o create a high-order model of a struc- ture, and then model reduction techniques are used to reduce the order of the model to a level appropriate for low-frequency analysis. This paper presents a method for beginning with a model of extremely high order and systematically eliminating degrees of freedom, but in such a way tha t they can be selectively retrieved to recover the model detail and refine the model. Since the method is developed with structural acoustic prob- lems in mind, in which the response is increasingly 1~ calized as frequency increases, it is important to be able to refine the model locally. This requirement sug- gests a substructure-level approach, which offers the added benefit tha t it lends itself naturally to paral-

Copyr~ght S 1992 Amencan Institute of .4eronautics and Artronaut~cs, Inc. .411 rights reserhed. 1631

lelization. The second section of the paper presents the first phase of this method, which is a multi-level substructure model reduction phase that yields a very low-order model of a structure. The next section de- scribes how this model can be optimally refined in the second phase of the method, which is a model refine- ment phase in which modes of substructures on various levels can be selectively included in the model to ob- tain optimal accuracy with a given number of degrees of freedom. The following section presents a numer- ical example which i l l~s t~ra tes the method. T h e final section contains conclusions.

"Bottom-Up" M o d e l Reduction and Assembly

This section describes a procedure for multi-level substructuring and model reduction, where degrees of freedom are eliminated in such a way that they can be efficiently retrieved to selectively enrich the model of the overall structure. A substructure a t the lowest level could be a single structural member, if such a rneniber is geometrically complex enough that it would be modeled as an assembly of finite elements. On the othcr hand, if structural members are each modeled with a single element, a substructure a t the lowest level would co~lsist of an assembly ofstructural members. In either case, the displacement degrees of freedom for the substructure are easily partitioned into those that are local, or internal, to the substructure, and those that are shared wit,h the rest of the structure at interfaces,

It is implicitly assumed that the finite element model associated with these degrees of freedom is adequate for achieving the desired level of accuracy in the anal- ysis of the overall structure. Finite element discretiza- tion results in substructure mass, damping and stiff- ness matrices which can be similarly partitioned:

L CLS = [CSL Css]

I' - I L L ~ C L S - [LL liss]

'The finite element representation is transformed into a new one which lends itself to assembly with neigh- boring substructures and selective inclusion of degrees of freedom. The new representation has two compo- nents, and the first component is associated with static response of the substructure to interaction with the rest of the structure through the interfaces. This is governed by the equation

where fs contains forces applied to the substruc- ture through interfaces. For this problem, the re- sponse in internal degrees of freedom UL can be ob- tained in terms of interface degrees of freedom u s , using the upper partition of the equation above, as UL = - K ; ~ K ~ S U ~ , SO that the representation

spans the static response of the substructure when no internal excitation is applied, and allows for arbitrary u s . Each column of the matrix Q represents one of the so-called "constraint modes" in the component mode synthesis literature516 and contains coefficients of finite element interpolation functions, so that a new interpo- lation function is defined throughout the substructure in terms of finite element interpolation functions. If conforming elements are used to model the structure, the displacement degrees of freedom represented by Q constitute a minimal set for guaranteeing exact com- patibility with adjacent substructure^.^ I t is not pos- sible, however, to represent arbitrary displacements in local degrees of freedom in terms of the matrix Q.

To complete the basis, a second component of the internal displacement UL can be written as aLq, where the vector 77 contains coefficients of the vectors in iDL, and QL is the square matrix that satisfies the algebraic eigenvalue problem K L L @ ~ = M L L @ ~ A . Here A is a diagonal matrix of eigenvalues, which are squares of natural frequencies. Each vector in QL contains c~ efficients of the finite element interpolation functions for approximating the substructure's natural modes of vibration with fixed interfaces, so the vector 77 con- tains modal coordinates for the substructure. The use of modes for completing the basis for the sub- structure is attractive because they constitute a set of displacement degrees of freedom that are ordered in frequency, and their orthogonality properties result in diagonal partitions of the substructure mass and stiffness matrices. The question of what type of com-' ponent vibration modes t o use amounts to a question of what boundary conditions should be specified for the substructure. It will be impossible in general to predict the interface behavior for the substructure in the response of the structure, and for complex three- dimensional structures, the modal density is usually high enough that individual modes are not dominant in the response as they are for simple problems. With these considerations in mind, i t is reasonable t o use fixed-interface boundary conditions for computational convenience, and these result in the algebraic eigen- value problem given earlier in this paragraph. It will be assumed that the vectors in QL are normalized so that @:M~L@L = I and @ : I ~ L ~ @ L = A , where I is the identity matrix.

With the above definitions, an arbitrary displace-

ment vector for the substructure can be represented as

in which the 9 submatrix can represent arbitrary in- terface displacement and can therefore handle com- patibility with adjacent substructures, and the a sub- matrix contains a frequency-ordered complement for Q with no influence on compatibility, from which specific modes can be selected for inclusion in the substructure model. The mass, damping and stiffness matrices for the substructure are transformed for this representa- tion t o

The fact that the off-diagonal partitions in the trans- formed stiffness matrix are null is easily verified and contributes to being able to accurately and economi- cally estimate how substructure models should be re- fined most ~fficient~ly, as will be seen in the next section of the paper.

It should be rernarked that representing the dis- placement u in terms of the vectors in 9 and is anal- ogous to using the hierarchical finite element method in one dimension. The vectors in Q are analogous t o basis functions for an element, which must be chosen so that compatibility with adjacent elements is assured. The vectors in are analogous to hierarchical func- tions, which can be added to the element to enrich the model, but which have no impact on compatibility, a t least in one dimension.

Once the transformation from the original finite el- ement basis to the {@, Q } basis has been accomplished, the transformed mass, damping and stiffness matrices can be assembled with those for other substructures according to the usual finite element assembly pro- cedure. As an example to illustrate the assembly of substructure models, consider a "subassembly" con- sisting of two adjacent substructures that share all of the nodal displacements in ui and ui. Compatibility between them is enforced by setting uk = ui us. Then arbitrary displacement for the subassembly can be represented by

where the complete submatrix Q is obtained from Q1 and Q2 by insisting on equality of interface degrees of freedom. Mass, damping and stiffness matrices for the two-substructure subassembly take the form

so that matrices for substructures assemble like ele- ment matrices, with a simple addition of the portions associated with common degrees of freedom. Note that modes of vibration for the substructures can be in- cluded or deleted from the model by simply including or deleting the corresponding entries in the v1 and v2 vectors, and the corresponding rows and columns in the assembled matrices. In practice, subassemblies will consist of a number of contiguous substructures.

Once a subassembly model has been assembled, it can be identified as a substructure a t the next higher level. Degrees of freedom that are shared with the rest of the structure are identified, and the remaining de- grees of freedom that were shared between lower level substructures are treated as the higher level substruc- ture's local degrees of freedom. Then the basis for the higher level substructure's displacement is trans- formed from being explicitly in terms of its shared and local degrees of freedom, to being in terms of Q and matrices obtained in the same manner as for the lower level substructures. In this transformation procedure, the local degrees of freedom for lower level substruc- tures do not appear explicitly, but are expressed in terms of shared degrees of freedom by means of the ?Ir matrices for lower level substructures. Upon com- pletion of the transformation, there will be modes for both the lower level substructures and the higher level substructure. Again, the transformed model of the higher level substructure is assembled together with other substructures a t the same level to form a sub- structure a t the next higher level, and the process is repeated until a model for the entire structure is ob- tained.

I t is advantageous for the purpose of minimizing matrix profiles to order the modes for substructures in such a way that the modes for lowest level "child" substructures having the same "parent" are concate- nated, and then followed by the modes for the "par- ent" substructure. Then the modes for this "parent"

Figure I : Matrix profiles for (a) modal damping, (b) general damping

substructure and its "children" are followed by modes for "children" of an adjacent (sibling) "parent" sub- structure. h4odes for all "cliildren" and "parents" un- der the same "grandparent" substructure are similarly collected, and all of tllese are followed by modes of the "grandparent" substructure. This pattern is con- tinued until the modes for all substructures a t all lev- els are included. Figure 1 shows the resulting pro- files for the matrix A ( w ) f -w2M + iwC + I< for a four-level decomposition of a structure, for modal and general damping. The off-diagonal rectangular blocks are fully populated and contain the coupling between a "parent" substructure and its "children" from the

transforlned mass and damping matrices The data is arranged efficiently for storage and for solution with a profile solver. If a mode for a substructure a t some level is to be deleted from the structurr rnodel, the cor- responding row and colurnn are s m p l y deleted from the mass, damping and stiffness matrices. Conversely, modes can be added t o the model by retrieving deleted rows and columns.

T h e important feature of a structure model in the form described here is t ha t since modes for components a t any level can be sclectively added to the model to improve accuracy, the model reduction and assembly process establishes a path for top-down refinement of the model down to the lowest level at which detail is important t o the accuracy of the solution. T h e finite element method as it is conventionally used defines lo- cal interpolation functions within individual elements, but in the method described here it provides a means for defining interpolation functions over much larger regions corresponding to st,ructural components a t var- ious levels, where these functions are entirely consis- tent with a detailed description of the geo~rietric con- figuration of the structure. T h e question that must be addressed, given this rnethod of ~iiodel reduction which allows for selective refinement of the model, is how to determine the most efficient way to refine the model. This is the topic of the nes t section.

"Top-Down" Adaptive Model Refinement

This section explains how the improvement in accu- racy resulting from including individual modes of sub- structures in the model of a complex structure can be estimated, so tha t the model can be refined optimally to achieve a desired level of solution accuracy. With the assumption tha t the detailed finite element model of the structure accurately represents i ts behavior, the motion of the struct,ure in steady-state harmonic re- sponse is governed by the equation

where w is the frequency of excitation and response, the matrices M , C, and K are the mass, damping and stiffness matrices for the detaiied finite element model of the structure, so they are of very high order, and the vectors u and F are response and excitation vectors, respectively. Typically, response over some frequency range, given some spatial distribution of excitation de- fined by F, is of interest. For convenience, the term eiUt will be dropped, and the frequency-dependent ma- trix A(w) r -w2M + iwC + I< will be introduced.

For determining how t o efficiently refine the re- duced structure model a t various frequencies, esti- mates of accuracy improvement from adding substruc- ture modes can be made using an approximate solution obtained with fewer modes in the model. The approx- imate solution before model refinement, upon which

estimates of accuracy iniprovemerit will be based, will be denoted by .CL, and the correction t o this solution resulting from including new substructure modes will be denoted by AIL. The exact solution to the equation given above, with all degrees of freedom included, will he denoted as u,,,,t. An crror vector will be defined as

e G ueXac t - ( f i + Au) . ( 1 0 )

A n error measure related to elastic potential energy is advantageous for estimating accuracy improvement, as will be seen, arid is defined by

where zT denotes the conjugate transpose of e . This error measure is a norm of the error unless the struc- ture possesses rigid-body modes, in which case it be- comes a semi-norm, because error in the response of the rigid-body modes does not. contribute to it. How- ever, this is inconsequential because the model reduc- tion approach of the preceding section preserves the rigid-body modes exactly in any reduced structure model, so tha t if they are included in the structure rnodel there will be no error in their response. The advantage of using the stiffness matrix in the defini- tlon of the error rrlemure comes from the ort!logonality of the substructure modes, w l ~ i c l ~ will permit estimates of accuracy irnproverricnt resulting from model refine- ment to be done independently for different substruc- ture modes. Expansion of the error measure results in the following:

iVithin a subst,ructnre, the equations of motion take the form

where F L and F s contain external forces applied to the subs tn~c tu re and f s represents the force ex- erted on this substructure by adjacent substructures through the interface. As a result of using the two-part substructure representation described in the preceding section, in which the displacement is represented as

these equations become

The coupling between the portion of the solutiori as- sociated with coefficients in u s and the modal portion of the solution is represented in the off-diagonal par- titions of the matrix above, and this coupling varies with w . Note tha t there is no term in these partitions without a coefficient of w because Kss vanishes, as mentioned in the preceding section.

If the approximate solution u has been obtained, U S and i j will be available, and i j will have entries of zero for modes tha t were not included in the model for u. For estimating how much the accuracy will be im- proved by including substructure modes in the model, i t will be necessary t o have a n estimate of the coef- ficients of substructure modes that may be included. The exact solution for 77 can be approximated by tak- ing us t o be correct, in which case the upper equations above can be written

where it has been recalled that MQu = (PTM9 and Caq = aTCQ. With the additional assumption that off-diagonal terms in C+@, i.e., modal coupling terms due to damping, can be neglected, these equations be- come uncoupled, and the one for the r th mode takes the form

where c,, is the r t h diagonal entry in CQQ, w, is the r t h natural frequency, and 4, is the r th eigenvector in (PL. Note tha t vectors mT ~ T [ M ~ ~ MLS]Q and cT 4T[cLL CLs]Q can be obtained ahead of time for each substructure mode, a t the same time that the transformation from the finite element basis to the { * , @ I basis is done, for multiplication by u s a t any frequency.

If this method is to be employed in a frequency sweep analysis of a structure starting a t zero frequency, modes for which w, is much larger than w can be ex- pected t o have negligible participation in the solution. As w increases t o the point where the participation of the r t h mode begins to become significant, the above equation can be solved to obtain a reasonable approx- imation for 7,. Obtaining 7, exactly would require correcting us , which would require solving the prob- lem for the whole structure again because of coupling with other substructures, and it is obviously impracti- cal t o do this for each of the substructure modes. How- ever, the correction to u s resulting from changing 7,. from zero to this finite value will be very small while 7, is small. In fact, the process leading to this first approximation for 7, can be seen as the first cycle of an iterative scheme (specifically, a block Gauss-Seidel

scheme) in which values for the degrees of freedom cur- rently included in the model and for q, are alternately updated. While q, is still small, this scheme converges rapidly, so that us and the value for q, obtained us- ing us can be regarded as the first terms in rapidly converging series, and using them as approximations amounts to truncating higher-order terms in the series. Recall that the objective here is t o estimate accuracy improvements resulting from including individual sub- structure modes, rather than t o obtain a solution, so the consequences of making this approximation can be expected t o be minimal.

According t o this approximation, u = uex,,t in degrees of freedom already included in the model, and including substructure modes in the model, with coeffi- cients satisfying the last equation, will reduce the error in the approximate solution u, and eliminate it com- pletely if all substructure modes are included. With the error measure given as

the first term represents the error in the approxi- mate solution u, which is independent of the cor- rection A u . Within a substructure, A u is repre- sented as A u = QAq, according to this approxima- tion, with nonzero entries in AT/ corresponding only to new substructure modes included in the model, and ueXact - u = @(qexact - 6) . The reduction in the error measure due t o the correction A u , within one substructure, is then approximated by

but since

this is simply

where A, = W: is the r t h eigenvalue for the substruc- ture. In this sum, Aq, is nonzero only for substruc- ture modes that are newly added to the model. Also, qr,exact - 77,. = Aq, for these modes. Hence, if S is the set of substructure modes to be added to the model, the reduction in the error measure for this substruc- ture is simply

For the entire structure, the reduction in the error measure is approximated by a sum over all of the sub- structure modes t o be added t o the model, for sub- structures a t all levels, of the squares of the moduli of the new modal coordinates, multiplied by the asso- ciated eigenvalues, so that all modes contribute inde- pendently to the reduction of Ilellk.

Once the improvement in accuracy is estimated for each mode of each substructure, all of the substructure modes can be ranked in order of how much including them would improve the accuracy of the analysis. The required accuracy of the analysis can be specified a priori, e.g., in the form

where 6 is an error limit specified by the analyst, and I I u I I ~ is available from a sum over the substructure modes similar to the sum for Ilell&. Then determining how to refine the model most efficiently to achieve this level of accuracy becomes a matter of simply including the substructure modes in rank order until the accu- racy improvement sum over the remaining substruc- ture modes is sufficiently small t o satisfy the accuracy requirement.

The natural order for making accuracy improve- ment estimates for substructure modes is in order of natural frequency. This suggests that if excitation is a t low frequency, estimates should be made first for substructures a t the highest level, since these substruc- tures have the lowest natural frequencies. As modes with higher natural frequencies are examined, it will be necessary t o check the modes of substructures on progressively lower levels. Accuracy improvement es- timates must be made until it is evident that accuracy is unlikely t o be improved significantly by checking modes higher in frequency.

Numerical Example The method presented in the preceding sections has

been applied t o a simple problem consisting of a uni- form Bernoulli-Euler beam that is fixed a t both ends and is subject to a localized harmonic excitation. This beam is shown in Fig. 2. The steady-state transverse vibration v(x, t ) of the beam is governed by the partial differential equation

where m is the mass per unit length, EI is the flexu- ral rigidity, y is a structural damping loss factor, set equal to 0.004, F is the force amplitude, w is the ex- citation frequency, and f is the excitation location, equal t o 0.7L in this example. The coefficient c is for distributed viscous damping, which is artificially

Figure 2: Beam for numerical example

introduced into the problem to localize the response around the excitation, so that the performance of the method can be investigated for cases in which response is localized.

The exact solution for this problem is readily ob- tained in terms of complex exponentials in x , and is shown in Fig. 3 for w = 3 2 , 0 0 0 J m and two values of c . For the first of these, the response is sig- nificant over the length of the beam. For the second value of c, the response is significant only over about twenty percent of the beam.

The beam is modeled using a multi-level substruc- turing approach, so that it is divided into four sub- structures a t the top level, identified as Level 0 in Fig. 2. Each of these substructures is subdivided into four substructures on Level 1, which are each recur- sively subdivided for lower levels. For the results pre- sented here, seven levels were used (Levels 0-6)) and a t the lowest level the beam is modeled in terms of cubic beam elements. T h e overall model has a total of 32,766 degrees of freedom. When the adaptive multi- level substructuring method is used in the analysis of the beam, convergence in terms of the ratio between Ilell$ and Iluezact / I $ , where ueZact is the solution of the problem with 32,766 degrees of freedom, is shown in Fig. 4 as a function of the number of degrees of free- dom for up to about two thousand degrees of freedom. The dashed lines are the results obtained with uni- form refinement of the mesh by the h-version of the fi- nite element method. The improvement in accuracy is particularly dramatic for the second case in which the response is more localized. In the asymptotic range, convergence is described by the equation

in which N is the number of degrees of freedom. For the first case, C z 1.9 x lo6 and cu !e 4.4, and for the second case, C z 2.5 x lo6 and cu z 5.0. For the uniform h-version, C w 7.6 x lo5 and cu z 3.4 for the first case, and for the second case, C z 4.4 x 10' and a z 3.0. This results in being able to achieve the same accuracy with a t least an order of magnitude fewer degrees of freedom with the adaptive method

Figure 3: Exact solutions: (a) c = ~ o o o [ ~ E I / L ~ ] ' / ~ , (b) c = 6 4 0 0 0 [ m ~ I / ~ ~ ] ~ / ~

when the response is localized. As would be expected, the benefits of an adaptive method are reduced when the response extends over the whole domain, although they are still significant. In one respect, this exam- ple problem is not representative of realistic problems because the motion is restricted to flexural vibration in a single direction, whereas in many problems the response is three-dimensional.

In a frequency sweep analysis, less computation would be required if the estimates of how much each mode contributes to the accuracy could be made in- frequently as the frequency increases. This is the case if modal contributions t o accuracy vary smoothly with frequency. Figure 5 shows how they vary over a range in frequency, where each curve is associated with a dif- ferent substructure mode. T h e first plot shows curves

Degrees of freedom

Degrees of freedom

Figure 4: Convergence for (a) c = 4 0 0 0 [ ~ 1 / r n ~ ~ ] ~ / ~ , (h) c = 6 4 0 0 0 [ ~ 1 / r n ~ ~ ] ' / ~

for modes contributing up to 100% of 1/u/1$, while the second and third plots show curves for modes con- tributing in the ranges 0-10% and 0-1%, respectively. The smoothness of the curves indicates tha t the esti- mates of which modes should be included in the model can be spaced rather far apart in frequency, possibly using interpolation between estimates to schedule the inclusion of modes in a frequency sweep analysis.

C o n c l u s i o n s

In this paper, a method for automated adaptive modeling of complex structures which optimizes the accuracy obtained with a given number of degrees of freedom is presented. T h e method uses a multi-level

Figure 5: Modal contributions t o accuracy in the ranges (a) 0-loo%, (b) 0-10%) (c) 0-1%

substructuring approach, in which the structure is par- titioned into several substructures a t the top level, each of which is itself divided in to substructures a t a lower level, and so on. This approach facilitates efficient representation of localized behavior. I t also lends itself t o parallelization. A numerical example is presented which shows that convergence is signifi- cantly faster than when uniform refinement is used, especially when the response is localized. T h e modal contributions to the accuracy of the solution are found to vary smoothly with frequency, which indicates tha t

the estimates used t o determine which modes should be included in the model can be made illfrequently in the frequcr~cy spectrum.

Acknowledgenient

This research is sponsored by the Department of the Navy, Office of the Chief of Naval Research, under grant number N0001491-J-1914. The U S . Govern- ment is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copy- right notation thereon. This manuscript is submitted for publication with the understanding that the U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessar- ily representing the official policies or endorsements, either express or implied, of the U.S. Government.

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