8.8 A COMMENTARY ()NCOMI~UTATIONAL MECl-IANICS*

J. T. ODENThe University of Texas at Austin

Austin, Texas

K. J. BATHEMassachusetts Institute of Technology

Cambridge, Massachusetts

INTRODUCTI ON

At a recent national scientific meeting, a panel of expertsdiscussed trends in computational methods in mechanics andthe physical sciences. One expert, eager to say something pro-found to get the discussion off on the right track, said some·thing to the effect that "Within the next decade, the only useaerodynamicists will have for wind tunnels is as a place to storecomputer print'out," This statement reflects two importantphenomena that have fallen on the scientific community inthe past two decades-the first is physical, political and soci-ological, and the second is psychological.

The first of these has to do with the significant advancesthat have been made during recent years in computing ma-chinery, numerical methods, and mathematical modeling. Itis an undeniable fact that, in many areas of technology, con-fidence in computer modeling has put an end to some of theexpensive laboratory testing that was so essential a few yearsago.

The second thought that our expert's statement provokesis the blatant overconfidence, indeed the arrogance, of manyworking in the field of computational mechanics. Some referto this point of view as "the number crunching syndrome"and warn that it is becoming a disease of epidemic proportionsin the computational mechanics community. Acute symptomsare the naive viewpoint that because gargantuan computers arenow available, one can code all the complicated equations ofphysics, grind out some numbers, and thereby describe everyphysical phenomena of interest to mankind. All the major

"Reprinted from Applied Mechanics Reviews.Vol. 31, No.8, August, 1978.

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problems of theoretical mechanics can now (or soon will be)solved on the computer.

Our aim here is to discuss both of these phenomena toamplify on the first, and to describe some trends in researchand education in mechanics and computational methods whichpromote the second. We want to point out that computationalmechanics is having an enormous impact on our lives, morethan many people realize, and that the benefits of recent ad-vances are important and far reaching. Secondly, we want topoint out that computational mechanics can be a "good" sci-ence. In the spirit of Shaw, it uncovers more questions than itanswers. It is also an intellectually rich and challenging subject.and an important addition to the family of subjects compris-ing theoretical and applied mechanics. Finally, we want topoint out that there are formidable open problems awaitingthe creative scientific mind in this field, and that even if somesubstantial breakthroughs were made, the confidence exhibitedby our expert is still not justified.

COMPUTATIONAL MECHANICS

The advent of electronic computation has had a profoundeffect on theoretical and appl ied mechanics during the lasttwo decades. The use of computers has transformed much oftheoretical mechanics into a practical tool and an essentialbasis for a multitude of technological developments whicheffect large facets of our everyday life. Indeed, in the world'sindustrial nations, it is almost impossible to go about one'snormal activities without coming into contact with some prod-uct that has been directly or indirectly the result of computerapplications of the principles of mechanics.

There has emerged from the work of recent decades a newdiscipline, lying essentially in the intersection of theoretical

mechanics, computer science, and numerical analysis, whichwe choose to call computational mechanics. It draws fromdevelopments made in each of these disciplines and is nour-ished by them, but it also stands as a viable branch of knowl-edge in its own right.

There are four basic components of the field of computa-tional mechanics:

1. Theoretical mechanics: the theoretical study of themechanical behavior of physical systems and bodies.

2. Approximation theory: the formulation of discretemodels, which are approximations of the equations of me-chanics.

3. Numerical analysis: the development of numericalmethods to analyze these models.

4. Software development: the development of computersoftware to implement the numerical methods.

In addition, one might add to this list, (5) Computer hard-ware: the development of computing machines to implementthe software. This aspect of computational mechanics is largelyin the hands of computer architects who have little interest inmechanics or its applications, Moreover, there is little to sayabout this subject that has not already been said many times-the development of sophisticated computer hardware has con·tinued to grow at an exponential rate since the late 1950's.There is every reason to believe that this trend will continue;we will not comment further on trends or the status of com-puter hardware development or research. The comments wemake here pertain to items 1 through 4, and particularly items1,2, and 3.

Our aim is not so much to summarize existing numericalmethods, approximation theories, or the status of theoreticalmechanics, but rather to describe what can and is being donein these fields today and point to some obstacles now standingin the way of future developments.

We first must recognize that there are two distinct aspectsof computational mechanics, just as there are in mechanics it-self. First, there is the mission-oriented and application area;that is the applied or engineering computational mechanicsdiscipline. The objective here is a practical one: to developcomputational techniques that aid in the numerical analysisof various models of physical behavior and to use these analy-ses to design actual mechanical systems-structures, machines,processes, etc. Usually, the final product of these efforts is acomputer program.

Second, there is basic computational mechanics research.Here the goals are to understand the fundamental propertiesof approximations and of the equations of mechanics, to ana-lyze numerical methods appropriate to study them, and tostudy techniques for optimizing software to implement thesemethods.

The contrast between these two areas is analogous to thatwe see in Numerical analysis, as opposed to numerical Analysisthe first aims to study the behavior of solutions to mathemat-

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ical problems, particularly differential equations, numerically;the second aims at using the techniques of mathematical analy-sis to study properties of numerical methods. A similar com-parison can be made between engineering mechanics and me-chanics. In basic computational mechanics, it is impossible(or, if not impossible, at best foolhardy) to separate a studyof the mechanical theories describing the phenomena at handfrom approximation theory. Physical arguments traditionallyhave been a guide to the development of sound mathematicalmodels; mathematics (for example, existence theory) andphysical experiments are final tests of the validity of the the-ory. Both are needed to formulate sound discrete models andboth provide a starting point for numerical analysis.

It is well known, of course, that what is research to oneperson may be development (i.e., the assimilation and appli-cation of known facts) to another, but we are concerned thatmuch of the activity now being labelled as research in compu-tational methods is, in fact, a sophisticated kind of develop-mental endeavor which is well outside the realm of good sci-ence or mathematics and, in the long run, cannot lead to im-portant scientific achievements.

APPLIED COMPUTATIONAL MECHANICS

The brief history of applied computational mechanics iscertainly a remarkable success story. Emerging from the em·piricism and crude approximate mechanical theories prevalentin the late 1950's, the subject has now reached a level of de·velopment where it affects not only the lives of everyone inthe scientific community, but also the.public at large. Newjournals on applied numerical analysis appear at increasingrates. International conferences, NATO meetings, symposia,colloquia, are held somewhere virtually every week of theyear. International commissions and organizations, some fullysupported and representing important industrial nations, havebeen organized on numerical methods and software. The ele-ments of programming are now being taught to high schoolstudents, and every college graduate in engineering and thephysical sciences is now expected to know some numericalanalysis and programming. A multitude of private companieshave appeared which make their livelihood selling computersoftware for the analysis of engineering problems.

To further emphasize the point, let us list a few examplesof the breadth of computational mechanics in our country:

1. Computational procedures implemented in computerprograms are employed throughout most of the nation's largermanufacturing companies on an everyday basis in the analysisand design of structures and mechanical equipment. Many ofthe larger companies have installed computer programs ontheir own in-house computers and organizations with smallerresources now have access to computer programs on a royaltybasis furnished by a large number of commercial computingcenters.

2. Every major nuclear industry, and especially all of thenuclear energy laboratories of the federal government, areheavily involved in computer applications of solid, structural,and fluid mechanics in the design of nuclear reactors. Thesafety and structural integrity of these reactors is a criticalaspect of their design and probably could not have been ac-complished had there not been significant advances in compu-tational methods during the last ten years.

3. The major automobile manufacturers in the world nowuse large scale computer programs as standard operating pro-cedure for the stress analysis, structural design, and dynamicanalysis of motor vehicles. The direct application of new ideasin computational mechanics has resulted in very large savingsin the auto industry.

4. A large portion of the nation's hardware and softwaredevelopment for national defense depends directly on com-putational methods for mechanics problems. The appl icationsrange over such studies as interior, exterior, and terminal bal-listics of projectiles; aerodynamics; penetration and impactstudies; control of missiles and aircraft; ablation of metalsunder aerodynamic heating; fracture; the overall structuralintegrity of water and airborne weapons systems; and satellitedynamics and control.

5. Since the early 1960's, the _computer-based analysis ofthe mechanical behavior of commercial and military aircraft,ocean liners, oil tankers, and high speed railway systems hasbeen an essential aspect of their design, Without reliable com-putational techniques, high speed computers, and a numberof recent technological advances, the current level of aircrafttechnology we enjoy would not be possible.

6. Virtually every aspect of our nation's space programhas relied heavily on computer applications of mechanicalprinciples. These range from the stress and stabilitY analysisof the Saturn V rocket and its predecessors, the aerodynamicsof space vehicles, the mechanical behaviors of fuels, and theanalyses of various material properties to the calcul ation ofoptimal orbits and vehicle dynamics of missiles and artificialsatellites.

To this Iist, we could add the heavy use of large scale com-puters in analyzing modern hydrodynamic models of the loweratmosphere and of the sea to study and predict the weatherand the flow of ocean currents; also, problems in water andair pollution are studied using complex models of the motionof rivers, streams, and harbors and of the transmission of pol-lutants through the atmosphere. We could also add the relianceof the petrochemical industry on computer modeling to simu-late oil and gas reservoirs, estimate their capacities, and designrecovery methods and storage facilities.

Additional activities could be mentioned, but the essentialpoint is that applied computational mechanics is now an in-trinsic part of everyday life in the industrialized world. HUn-dreds of millions of dollars are invested annually in computa-tional activities of the types described, and these have been

very important in providing the high standard of living we en-joy today. More importantly, the continued development ofbetter computational capabilities and more encompassing me-chanical theories is essential to our maintaining this standardin the future.

EXPERIMENTAL AND COMPUTATIONAL MECHANICS

Let us return to the question of computational methods asa substitute for physical experiments. During the short span ofyears in which computational methods developed into an im-portant element of theoretical mechanics, experimental me-chanics was by no means dormant-it also enjoyed a periodof growth and development unequalled in its history. Moderncomputing equipment and numerical modeling have broughtimportant new tools to the experimentalist, just as physicalexperiments and measurements have provided one of the fewrational means for testing the validity of many of the compu-tations which led to the achievements listed in the previoussection. Modern experimental and computational methods,combined in an interactive way in which one complements theother; represent some of the most powerful tools available forstudying many complex physical phenomena. The use of suchtools in characterizing materials; testing the validity of me-chanical and thermomechanical theories; verifying and evalu-ating numerical models and their underlying assumptions andlimitations; and as a guide to resolve such questions as "at whatlocation in a system and at what time should measurementsbe taken" can have a significant impact on basic and appliedmechanics in years to come.

AN ERA OF COMPUTATIONAL EMPIRICISM

How rei iable are numerical experiments as a means fortesting the effectiveness of numerical methods? We are re-minded of the numerical experiments on nonconforming finiteelements in the 1960's. By solving numerically a large collec-tion of sample problems, one group of analysts concluded thattheir element was "very competitive" as an element for theanalysis of plate bending problems. It is now known that thatelement is divergent! Their deceptively good results arose fromparticular choices of loading and boundary conditions which,because of special symmetries in the solutions, effectivelymasked the inadequacy of their element. Then there are theexamples of mixed finite elements in the late 1960's, whichare numerically inconsistent, but which could produce resultswhich looked reasonable for a specific mesh.

There is also the example of a well-known shell analysiscode circulated in the late 1960's and early 1970's and usedthroughout the United States which employs and inconsistent,and in fact divergent, shell element. Fortunately, many of theapplications of this code were accompanied by full scale labor-

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atory experiments which finally were the overriding factor indetermining designs of various structural systems.

The picture in the calculation of flow problems is not muchbetter. How rei iable are the large hydrodynamic codes? Theiraccuracy is often established by comparisons with experimentsor with special cases for which analytical solutions are known.In the difficult application areas described earlier, the idealconditions one has in those problems for which analytical solu-tions are known are practically never present.

In the engineering community of 20 years back, it was un-derstood that the use of classical analytical methods providedvery limited tools for studying mechanical behavior and, as aconsequence, it was mandatory for the engineer to supplementhis analyses with a great deal of judgment and intuition ac-cumulated over years of experience. Empiricism played a greatrole in engineering design; while some general theories of me-chanical behavior were available, methods for applying themwere still under development, and it was necessary to fall backupon approximation schemes and data taken from numeroustests and experiments.

Today it is a widely held view that the advent of electroniccomputation has put an end to this semiempirical era of engi-neering technology: nowadays, sophisticated mathematicalmodels can be constructed of some of the most complex phys-ical phenomena and, given a sufficiently large computer bud-get, numerical results can be produced which are believed togive some indication of the response of the system under in-vestigation.

In the case of linear problems, we now have a relativelybroad experience in computing solutions, In general, thereliability of the computed results is 5eldom questioned as ithas been corroborated by intuition, observation, and practicalexperimentation over a significant period of time. Moreover,error estimates are now available for many of the numericalmethods in use and these provide both a priori and a posteriorichecks on solution accuracy. It is certainly true that substan-tia' improvements in design, some of which were outlined inthe previous section, have resulted from confidence in the useof computers and numerical methods in recent times. Thetrend today is to continue to press the capabilities of compu-tational methods into areas of nonlinear mechanics in thehope of gaining still further improvements and a better under-standing of the response of actual engineering systems.

Unfortunately, in the case of nonlinear problems, fewnumerical experiments can be supplemented with judgmentgained by years of experience. Even worse, many computa-tional empiricists seem to be unaware of the fallibility of theirmethods. We believe that in the area of nonlinear mechanics,we have, ironically, left one semiempirical era and enteredanother-the empiricism now is manifesting itself in computa-tions.

There are two major difficulties inherent in the addition ofnonlinear analysis capability to the tools of contemporary

engineering analysis and design. The first is well known: anumerical analysis of a large scale nonlinear problem is ex-pensive, often 10 to 100 fold that of a comparable linearanalysis. Consequently, the choice of an appropriate techniqueand its effective implementation becomes critically importantin the practical analysis of nonlinear problems, Also, for largescale nonlinear problems, few institutions can afford to followthe practice common in linear analyses: solve the prOblem forseveral different meshes to test convergence; experiment onthe behavior of the solution to changes in material constantsand boundary conditions; investigate several loading condi-tions; and alter the geometry of the problems and comparespecial cases with known solutions when available. It is rarethat this type of numerical experimentation can be done effec-tively for nonlinear problems.

Also, considering the fact that computational methods areused as a basis for many important decisions which affectour security and livelihood, it is certainly unwise to employmethods whose reliability is not completely verified. Thisreliability of methods for nonlinear problems is the secondmajor difficulty that faces the developers of nonl inear codes.It makes little sense to invest hours of computing time on acomplex nonlinear analysis, only to obtain results which mayhave little bearing on the actual response of the system athand. There are many frightening examples which reinforcethis point. The use of finite element methods, for example, inthe calculation of postbuckling behavior of nonlinear struc-tures can be very sensitive to mesh size-a refinement in whatappears to be a reasonable mesh can produce gross changes inthe postbuckling response, and the true behavior of the sys-tem may be quite different from that produced by what manywould consider a reasonable model. In nonlinear evolutionproblems, examples abound of cases in which a small perturba-tion of the initial data can produce enormous changes in theresponse of the system-yet numerical approximations of solu-tions of problems of this type are ground out daily (many arepublished) with little evidence that the analyst is aware thathis solution may not resemble in any way that experiencedby the actual system.

There is much evidence that we are, indeed, in an era ofcomputational empiricism. If we are to leave this era, we mustmake a conscious distinction between applied and basic com-putational mechanics and caution the army of enthusiasticusers of modern software that much in the way of fundamentalresearch remains to be done before the nonlinear theories ofmechanics, in general, can be used with confidence as a basisfor analysis and design.

BASIC RESEARCH IN COMPUTATIONAL MECHANICS-SOME OPEN QUESTIONS

Space does not permit us to list all of the important openquestions standi ng in the way of the advancement of basic

computational mechanics. The following partial list, however,gives an indication of the flavor of some of the formidabledifficulties that remain to be resolved:

1. A POSTERIORI ESTIMATES: ADAPTIVE MESH RE-FINEMENT STRATEGI ES

Is it possible to measure quantitatively the accuracy ofnumerical solution by means of testing the solution after ithas been calculated? A posteriori estimates for the simplestlinear one-dimensional boundary value problems have beenestablished on a rigorous basis, and it is not uncommon forcertain a posteriori estimates of accuracy to be used in largescale hydrodynamic calculations, and in the solution of largesystems of stiff Qrdinary differential equations. These, how-ever, are otten based on rather shaky mathematical grounds.The development of a posteriori error bounds for difficulttwo· and three-dimensional problems in solid and fluid me-chanics requires computational tools that are not yet available.Once such a posteriori estimates are available, it should notonly lend some confidence to some of the methods now inuse, but also provide an important element in improving solu-tion efficiency. Indeed, the availability of a posteriori esti-mates would make possible the development of adaptive meshrefinement schemes which could conceivably produce nearoptimum meshes for the analysis of various boundary valueproblems. It could solve an old problem: the generation ofthe best possible numerical solution for a given problem for agiven amount of computing time or dollar expenditure.

2. NONLINEAR ELASTICITY PROBLEMS

The numerical analysis of the equations of finite elasticityis in an embryonic stage. Many computer programs for suchproblems exist, but there is virtually nothing known abouttheir accuracy outside of results produced by numerical ex-periments. Error estimates and convergence criteria are knownfor some particular cases, but these are based on assumptionson the regularity and existence of solutions. Unfortunately,means for determining a priori the regularity of solutions formost nonlinear boundary value problems are not known. Evena local analysis of these equations leads to indefinite forms anddifficulties with nonuniqueness and instability of solutions. Acomplete numerical analysis of these problems lies beyond thecurrent state of the art.

We also mention that the key to the study of nonlinearboundary value problems, such as those encountered in finiteelasticity, is the availability of an existence theory. Indeed, anexistence theory not only provides the basis for an approxima-tion theory but it also provides the only tool outside of phys-ical arguments and laboratory experiments for establishing thevalidity of a theory itself. To date, a complete existence theoryfor finite elasticity does not exist and the development of an

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approximation theory must still be based on a long collectionof simplifying assumptions which mayor may not be valid forparticular problems of interest.

3. ELASTODYNAMICS

If the study of numerical analysis of the equations of non-linear elastostatics is not well established, then the state of theart in elastodynamic problems is even worse. Here it is knownthat smooth initial data may lead to the development of shockwaves, and that multiple solutions can exist. Moreover, not allthese solutions may be physically relevant, as positive entropymust be produced across a shock front. How can we modelsuch phenomena numerically? What means do we have to de-termine that a numerical scheme will not produce in someartificial wayan unrealistic solution-one that does have apositive entropy production across shock fronts, but which isonly due to some truncation error in the numerical process?Then there are the nonlinear vibration problems, completewith bifurcations and instabilities. How do we determine thesevarious solutions numerically? Which ones, if any, resemble thephysical behavior we wish to model, and what is their accu-racy?

4. INELASTIC MATERIALS

When we leave the realm of the mechanics of elastic mate-rials we enter a little explored area of continuum mechanicsin which little in the way of a complete mathematical theoryis known. Progress in, for example, the theory of finite plas-ticity has been made in recent years, but few would agree asto what constitutes a reasonable boundary value problem inthis area, much less a theory on the existence, uniqueness, andqualitative behavior of solutions of such problems. A mathe-matically sound approximation theory and numerical analysisof these problems cannot emerge until the problems them-selves are well set. One ~an add to these difficulties all of thecomputational and mathematical difficulties brought about inmodeling phase changes, viscous effects, cracks and singulari-ties, the growth of cracks under dynamic loading, and theidentification and implementation of physically reasonableconstitutive equations describing these materials:

5. STOKESIAN FLOWS

Existence and approximation theorems are available foronly certain cases of Stokesian flows. The few studies of errorand convergence of numerical methods for such classes ofproblems are typically based on truncation concepts whichare invalid for irreg~lar solutions. Bifurcation phenomena inthree-dimensional flows are only now being understood andthe study of the behavior of numerical approximations of

such phenomena is in an extremely early stage of develop-ment.

6. COMPRESSIBLE FLOW PROBLEMS

Perhaps some of the greatest difficulties in the way of de-veloping an approximation theory and numerical analysis areencountered in the study of compressible flows. The theoryof mixed nonlinear partial differential equations is very muchunderdeveloped. Theories on the qualitative behavior of solu-tions are known only for special cases and existence theoremsfor fully developed compressible flow problems are still farbeyond the methods of the modern theory of partial differen-tial equations. Since a rational numerical analysis must bebased on such theories, the likelihood that problems of thistype will soon be treated with numerical confidence is small.The picture is further complicated by the fact that the solutionof these equations may be very irregular, i.e., possess shocksand singularities. The development of schemes for handlingthese difficulties still poses major unsolved problems.

7. NON-NEWTONIAN FLOWS: CONVECTION-DIFFUSIONPROBLEMS

Many of the difficulties one has with Stokesian flows aremagnified in the case of non-Newtonian flows. Mathematicalbases for many models of non-Newtonian flows are not wellset. nor are criteria for constructing theories that will guaran-tee physically reasonable solutions of the governing equations.Some of the simpler fluids (e.g., Bingham fluids) can be char-acterized by equations for which there is a reasonably firmmathematical theory for certain choices of boundary and ini-tial conditions, but the development of a corresponding ap-proximation theory and numerical analysis is only in its earlystages. Convection-diffusion problems still present numericaldifficulties that have not been completely resolved-particu-larly in nonlinear convection-diffusion processes and/or incases in which free surfaces are developed which propagate intime.

CONCLUDING COMMENTS

The success of large scale computations has led to somecurious situations. As might be expected, it has had some im-pact on education in engineering and the physical sciences.Many of the general equations of nonlinear solid and fluidmechanics which were laid down in the 19th century are nowof more than purely academic interest. They must be under-stood, expanded, and generalized to provide a basis for model-ing many of the various phenomena descri bed earl ier. Thebreadth and scope of some of the applications listed previ-ously should di lute arguments once prevalent in our universi-

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ties that too much time is spent on the fundamental theoriesof continuum mechanics and that modern mathematics andnumerical analysis is a waste of time for students interested insolving real world problems. Thus, it would seem that the tri-umphs of applied computational mechanics would heightenthe appreciation of most practitioners for the need of morebasic work in all of the areas which feed such large scale com-putational efforts; however, with a few exceptions, the oppo-site has been true.

The trend in education, particularly in engineering schools,has definitely turned away from the more basic studies invogue during the late 1960's. In many schools, the basic equa-tions governing mechanical phenomena, if mentioned at all,are too often presented but not taught; emphasis is given tomethods to use as opposed to concepts on which to build; onhow to as opposed to why.

In many instances, this has produced technicians who prac-tice what some call computer overkill, which refers to the useof existing large sophisticated codes to study systems whichcould be better understood, at considerably less expense andeffort, by the use of much simpler models in the hands ofthose with a deeper comprehension of the physical phenom-ena and the basis of the methods available to analyze them.

In other instances, it has created the computerized giantkiller, who is ready to launch into the analysis of the mostcomplex physical problems and confidently provide "solu-tions" which may have significant implications in importanttechnological projects. Alexander Pope's classic line, "A littlelearning is a dangerous thing" was never more appropriate.

The reasons for such trends in education are complicated.They involve the increasing tendency of university administra-tions to manage their programs as businesses desi!Jled to com-pete for federal monies: their frequent adoption of short rangegoals results in training technologists to meet fluctuating in-dustrial needs rather than to provide a basis for learning.

Then there is also the attitude of some educators in engi-neering and the appl ied sciences that the phenomena theywish to study are just too complicated to ever warrant the useof modern theories or methods, and that one's only recourseis to reach backward to the semiempirical methods of decadespast, One hears this opinion voiced too frequently by some whowho are entrusted with the responsibility of directing and con-ducting basic research and who have some voice in determin-ing directions of educational programs. There is little newabout such positions; similar attitudes have always stood inthe way of progress in science and technology. In some re-spects this philosophy is diametrically opposite to that whichwe have emphasized herein-no confidence as opposed to over-confidence in modern computing capabilities-but it is equallydangerous, for it is not based on penetrating logic but ratheron ignorance of the scientific method and confusion as to thepurpose and responsibilities of institutions of higher learning.We hope it does not prevail.

ACKNOWLEDGMENT

We wish to thank Professors Richard H. Gallagher andMark Levinson, whose suggestions helped us improve certainsections of this commentary.

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