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A Computer System for the Analysis of Large Scale Problems in Solid, Fluid and Thermo-Dynamics
A COMPUTER SYSTEM FOR THE ANALYSIS OF LARGE SCALE PROBLEMS IN SOLID, FLUID AND THERMO - DYNAMICS
C. Carmignani Assistent . Professor Istituto di Meccanica Applicata Universita di Pisa Pisa, Italy
A. Cella Research Scientist Istituto Elaborazione dell'Informazione Consiglio Nazionale delle Ricerche Pisa, Italy
ABSTRACT
complex problems. SUCh30ftware is, usually, developed on a proprietary basis, by a limited number of industries and research institutions operating in the higly industri~ lized countries. The object of this paper is to outline the concepts beyond the development of a "home made" software package, such as the SELF s~ stem and to illustrate some of its achievements. The SELF system was developed in Pisa by a joint team from the National Council of Research (CNR) and from the Engineering Department of the University. Our .-,onclusion is that, when sufficent wisdom is placed in the initial decisions,such as: methodology, group organization, and release policy, the support of a relatively small research team can produce very rewarding results.
The technical design of large scale goverment projects often requires an elaborate computer analysis of different, mutually ig terfaced problems. The sophisticated computer software necessary for the analysis is often available only in the higly industri~ lized countries. The underdeveloped count-· ries, while sponsoring the projects, are often demoted from a number of important d~ cisions. The paper describes the concepts and the methods that lay beyond the develoE ment of a "home made" general purpose computer system, the SELF system, made operational in Pisa, Italy. The present achievments and the potential impact of the system are described. INTRODUCTION
A UNIFIED APPROACH A number of large scale government projects such as the construction of nuclear power plants, arch and gravity dams, pipelines, high rise building complexes, etc., face severe design problems, due to the size of the project, as well as to the interface needed between different disciplines, such as structural mechanics, fluid flows,thermQ dynamics, etc. The developing country while ultimately paying for the project, is often unable to perform the technical design of its own prQ jects, and resorts to the services of foreign consulting firms. Many decisions, then, are taken thousands of miles away from the site ,with limited feedback from the users of the facilities themselves. The technical gap that is at the base of such a distorted decision process is often the lack of access of the technical community in the developing country to the sophisticated software necessary for the computer analysis of large scale
The problems of mathematical physics are usually divided into four broad families: elliptic, eigenvalue, parabolic and hyperbolic problems. Without any attempt to rigor or to completeness one may typify the different problems as follows: elliptic: field equations (steady state heat flow, seepage in porous media, irrotational flow of inviscid fluids), elastic behaviour of structures and continua under static loads, etc. ; eigenvalue: vibrations, stability of structures, etc.; parabolic: diffusion equations, transient heat transfer, viscous fluid flows, etc. plastic behaviour of structures; hyperbolic: wave equations, elastic behaviour of structures under dynamic loads. In conventional engineering, each problem requires a particular computational technique, thus leading to specialized technical activities, concerned primarily with the
81
mixed parabolic and hyperbolic;
physical aspects of the problem and with limited crosscomunication. The recent revival of var~ational methods associated with the increasing use of large scale computers has set into motion a trend toward using a unified mathematica l model and a general purpose computer system (') ,('0 ). The most important contributions to the variational formulation of differential boundary value problems were given by Mikhlin (6) for elliptic and eigenvalue problems, and by Lions ( 5 ) for time-evolving problems. Quite independently, outstanding engineering schools put into operation powerful computer systems to solve a broad spectrum of civil and aeronautica l engineering problems ('},(9},(" . ) . S ubstantial research work is under progress (4)( 7 ) ( . }( 'O )('2) to achieve a unitary view of the formulation,approximation,and solutio n of the different Kinds of boundary value problems.
A ( t ) u ( t ) + B ( t ) u ' ( t ) + C(t }u" = f ( t ) In
n n
in
eigenvalue;
Au
AU
in
parabolic;
A (t ) u (t ) +u ' (t) = f( t} ; n , O
where
u"
in
o
a
2
n n
u
2
C (t) linear operators
a(u,n} = (Au,n)
n £V
The b ilin ear form is associated with the inner product,after the inte gration by parts is performed v} a linear functional is introduced, L(v} defined on V; vi}in problem (3) both sides of the equa tions are integrated between 0 and T, after performing the inner product.
(1 )
Problems (1},(2) and (3) are finally placed in the form see ( 4 ) , ( ~ ) , ( 6 ) ;
(2 )
a(u,.p} = L( cp}
(3 )
0
where
u'
in
o
O
Here the depende nce of the solution u from the "space" variables is implicit, while the dependence from the "time" parameter t is explicit. Similarly, the need for the solution u to satisfy " space " boundary conditions is cumulated in the r equirement for u to belong to the subspace D(A} of H whil e the initial conditions are explici~ 'I~e differential problems (1},(2},(3) are transformed with the following operations; i} both sides of the equations are innerly multiplied by a "test" function n belonging to HA; ii}the inner product (Au ,n), implying an in tegration ove r derivatives, undergoes an integration by parts, that lowers the or der of the derivatives a n d introduces the boundary conditions; iii}a subset V of the function space HA is taken into account, where the functions have derivatives square summable up to a certain order ; that is referred to as a Sobolev space , and is proved to be den se in HA ( 6 ) ; iv}a bilinear form is introduced, a(u ,n} de fined in VxV, such that
HA the completion of the subspace D(A) with respect to the energy norm;. In the given n otat ion, the above mentioned problems are represented in the following compact form ; f
u(o}
B( t},
In the following, the problems so far presented will be introduced in the concise functional analysis notation. For sake of brevity, we do not carry all the prelimin~ ry mathematica l notation, leaving to the r~ ferences the precise definition of the individual problems. Let A be a linear differentia l operator , symmetric and of the elliptic type; H a Hilbert space of functions; D(A} the subspace of functions of H opera ted upon by A; U,v functions of D(A}, defined on an open , bounded subset n of the Euclidean space RN; (u,v) the inner product of u and v in H ; Ilull (u,v)~ the norm of u in H; [u,v] = (Au,vl the "energy" inner product IlulIA= [U,U]2 the energy norm of U;
Au
u
at
A UNIFIED MATHEMATICAL MODEL
elliptic ;
u(O}
n
(4 )
where ~ is a properly defined subset of V . The conditions for the existence and the uniqueness of the solution of problem (5) aregivenby(5} ( 6).
au at
82
The differential problem (4), on the other hand, is transformed by Galerkin's method. The function u(t) 1S assumed to be of the separable form u(t) = w(x)y(t)
x£R
is organized and structured in three different areas: 1. Partition of the procedure into several levels of operations; 2. assembly of the subroutines into modular blocks; 3. structuring and processing of data sets. A brief outline follows of the main features of the SELF system. Partition of the procedure. One of the most attractive features of syste~ is the capabi lity of solving different problems. In the overall procedure, the operations that are specific to each problem are inserted "in parallel" into the flow of operations that are common to all problem. The macrof l owchart for dynamic analysis(eq. (9)) is shown in fig.l. Each block there ig dicates a phase of the procedure that oper~ tes independently from the other ones. A phase is in turn partitioned into functions. Functions can be of a general kind, or specific to a problem. Finally, the functions are broken into a string of elementary matrix operations. Modular blocks. The subroutines are set up and collected into modules so to minimize the number of subroutines and to adjust the length of the module to the primary memory available. In order to reduce the number of subroutines, the elementary operations are collected in a ~ tility module that can be accessed from each phase and function. A number of subroutines are sufficently general to be usedin more than one function. The modules are dynamically a llocat ed into primary memory from the monitor program, a£ cording to the flow of the procedures. The modules over lap to each other. Within the phases, the general functions and the functions specific to the problem are collected into different modules; thus for each problems only the related modules are allocated. Data structure. The working data , on which is performed the bUlk of operations , are parametrized by the number of nodal variables, that changes with the type of problem and eventually from node to n ode. Thee major phases have a monitor program : that, anticipating the operations to be performed, shifts into primary memory the data to be processed. The purpose is to minimize the number of accesses to the slow secondary m~ mory: thus the data are retrieved in blocks that are as large as required for the completion of the operation, or, otherwise, as large as the primary memory available .
N
performing the same operations, i) to v) as before one obtains, using an analogous nota tion a(t;w,
=
L (t; cp)
(6 )
APPROXIMATION AND SOLUTION Problem (5) is approximate by a finite dimensional problem where the functions u and n are sUbstituted by piecewise polynomial functions. The procedure is known as the finite element method( " ). For the computational details, and for an appraisal of the er.ror bounds see ( 3 ) , ( 8 ) . Calling ~ the vector of values of u at a fi nite number of points, and ~, ~ the vectorn of first and second derivatives with respect to time , the approximating problem takes the following f orm, in matrix notation elliptic,parabolic
Mu
P
(7 )
eigenvalue
Mu
AU
(8 )
mixed parabolic and hyperbolic M~(t) +N~(t)+Q~(t)=P(t) By applying the Fourier transform to eq.(9) one obtains: (10) where u(w) is the Fourier transform of ~(t), p(w) i; the Fourier transform of p(t), and Q(w) is a vector of terms that take into ac count the initial and the final conditions on the t i me v a r i ab 1 e. E q . (1 0) to sol v e d for a discrete set of values of the frequency w. Applying to u(w) the inverse Fourier transform one finally obtains u(t). Direct and inverse (fast) Fourier tra~sforms are comp!:!. ted by the Cooley-Tukey algorithm.
A MODULAR COMPUTER SYSTEM The flow of operations in the SELF system
83
INPUT
'--71> -:.,
no
~~~2:s;~2J~~~tt!;l~~~~~~bf >
" '\
~l-'.~t--t'-~~+~.~ +-:.-1._. --r~r-----;'~.~k--------'i<:-:-i-;f:-k'i
for every Wj
f-
assembly of the matrix
M+iw.N-w 2 Q J
i
J
L
Set up boundary
conditions for eq.(ll) Fi g. 2 - Shielding structure oC 8. nuclear reactor core.
Solution of eq.(ll)
sto rage into file of the solution u(w.) -
J
for every loading
,
condition P.(t)
,
retrieve P.(t) from file
Fourier transform
,
T(P.(t))
,-
Fig.
Fig. 1 - Macro flowchart for dynamic structural analysis .
84
3 - Enclosing shell of a nuclear reactor.
APPLICATIONS
a compact 3000 ca rd package. With respect to similar, large scale finite element sy stems like NASTRAN , ASKA, STRUDL,the ratio between numbers of cards ranges between VIO to 1/ 30, while the ratio of analytic capabi lities goes from 0 . 6 to 1. The sUbstantial economy in the number of cards is due to s~ veral factors; in order of relevance: i) the SELF system was designed for modern large scale computers, with time-sharing modes of operation and hardware based "virtual" memory; the handling of data in and ut of . core is reduced to a minimum; ii)the SELF system, solves internally, a small number of general mathematica l pr2blems; the variety of physical problems is confined to the input and output of data; iii)the continuity of the team work ; it l S proved nowaday that changes in the team composition or in the supporting contract affect the developing program with uncon trolled mushrooming.
The SELF system was developed for research purpose, and in such form released for general use. The modular structure, however, is such that many problems of practical interest can be solved by introducing minor modifications or extensions to the bulk of the system. Some institutions,then, have adopted the policy of extending the SELF s~ stem by their own means to fit their problems and their computational facilities. Here are some applications collected from various users. In fig.2 the shielding struc ture of the core of a nuclear reactor is shown, in its section view, together with the mesh size used for the analysis. The temperature distribution in the body was computed first; then a thermoelastic analysis was performed on part of the structure. In fig.3 is the perspective view of the outer shell of a nuclear plant. The s hell is stiffened by a large box girder ring(no shown here),and was analyzed for stresses and deflections under several loading con ditions. Fig.4 shows the plan view of the roof of a new cathedral ana lysed for stresses under wind loads. The 150' diameter reinforced concrete roof is corrugated, stiffened by curved panels, and carries a 30' high slab. Fig.5 shows me fluid section of a plasma gun. Assuming ste~ dy flow, the velocity distribution in the fluid was computed using the fine partition in triangles shown in the figure. The collection of those few examples indic~ tes, beyond the details of the individual problems, the versatility of a computer system, such as the SELF s ystem, as a broad range problem solver.
Another relevant characteristic of the SELF system is its open box availability.The sol ving technique can be learned, and improve~ not just borrowed. The impact of such policy on education and applied re sea rch i s obvious . Public institutions, that often carry the burden of che cking and controlling complex projects, have now free access t o a relia ble "home made" computer packa g e. The conclusion derived here, after the dev~ lopment of the SELF system, are: i) a great importance is attached to the planning phase of the development:the s ~ lection of the mathematical model, the structure of the syste m, the direction ci the possible extensions, etc.; ii)the development should be phased in con di tions of high efficency: a compact teeJll, concentrated in one place, a modern, la~ ge scale computers, broad availability of computer time, continuity of the resear& effort.
EVALUATION AND CONCLUSIONS The development of the core of the SELF sy stem took two years of a five man team. A sUbstantial amount of initial planning highlighted the basic functions of the sistem and the interface with possible extensions of both the analytic capabilities of the pr2gram and more sophisticated input output ~ rations. Thus, any modification the individual user may want to introduce can be addro as a branch to the core of the system wi thout upsetting the structure of the program. The ind i vidual subroutines are optimi zed with respect to number of instructions and core occupancy; the programs are written in FORTRAN IV for simplicity of transfer from one computer to an o ther. The system is
When two such conditions are met, a relative ly small funding effort, combined with an appropriate release policy that favors'~all outs" of the basic research toward applied research, may produce very rewarding resuhs. The developing countries are often mislead, by the so called technological gap ,i nto tr~ ing to upgrate their entire technical comm~ nity. Such policy often leads to dispersal of energies and to discouragment in front of the fast paced advances in technology. The concentration of efforts, on the other hand,
85
Fig.
y
view of the
4 - Plan of a cathedral . roof
[cm]
18
O~-----------------------~==~~~~402±~~~to50--x~[cmJ o
. . to elements . . 5 - Plasma gun: partit10n 1n flg
86
is certainly a short term solution; but it may lead to reducing the gap in a number of important areas, with precious savings for the country. Time is thus bought for the fairly slow diffusion process of the knowhow, and confidence is gained in the possibility of producing original contributions.
(5) Lions J.L., "Equations differentielles operationelles et Problemes aux Limites' Springer, Berlin, (1961). (6) Mikhlin S.G. ,"Mathematical Physics, an Advanced Course". North Holland, Am s t er d am, (1970).
AC KNOWLED GI4ENT
(7) Strang G., Fix G., "A Fourier analysis of the finite element method~ To appear in the Prentice-Hall Mathematical series.
The present research was partly supported by grant n055 0 from NATO Scientific Affairs Division.
(8) Strang G., "Approximation in the finite element method". Num. Math., 19,2, (1972~ 8l-98.
REFERENCES (1) Argyris, J.H., "The impact of the digital c o mputer on the engineering sciences". Part I', Aero. J.Roy.Aero.Soc., Vol. 74,pp.13-41, 1970.
(9) Strudl I., "Internal logic manual". IBM Publ. n.360-D 16.2.015, White Plains, N.Y., (1968).
(2) Carmignani C., Cella A., Lami V., Bozzi R., "11 sistema SELF/2 per il me todo degli elementi finiti". I.E.I. Re port B72-6, (1972).
(lO)Whiteman J.R., Barnhill R.E., "Finite element methods for elliptic mixed bOu£ dary value problems containing singularities". Proceeding EQUADIFF 3, Brno, Czechoslovakia, (Au~.1972).
(3) Cella A., "Approximation techniques in the finite element method". To appear in the CISM Springer Series, Wien.
(ll)Zienkiewicz, "The finite element method in engineering sciences". Mc Graw-Hill, New York, (1971).
(4) Cella A., Morandi Cecchi M., "An extended theory for the finite element method". Proceedings of the InternatiQ. nal Conference on Variational Methods in Engineering. Southampton (Sept .1972)
(12)Zlamal M., "Some recent advances in the mathematics of finite elements". Int. Conf. on Mathematics of Finite Elements, Brunel University (April 1972).
87
C. Carmignani Assistent . Professor Istituto di Meccanica Applicata Universita di Pisa Pisa, Italy
A. Cella Research Scientist Istituto Elaborazione dell'Informazione Consiglio Nazionale delle Ricerche Pisa, Italy
ABSTRACT
complex problems. SUCh30ftware is, usually, developed on a proprietary basis, by a limited number of industries and research institutions operating in the higly industri~ lized countries. The object of this paper is to outline the concepts beyond the development of a "home made" software package, such as the SELF s~ stem and to illustrate some of its achievements. The SELF system was developed in Pisa by a joint team from the National Council of Research (CNR) and from the Engineering Department of the University. Our .-,onclusion is that, when sufficent wisdom is placed in the initial decisions,such as: methodology, group organization, and release policy, the support of a relatively small research team can produce very rewarding results.
The technical design of large scale goverment projects often requires an elaborate computer analysis of different, mutually ig terfaced problems. The sophisticated computer software necessary for the analysis is often available only in the higly industri~ lized countries. The underdeveloped count-· ries, while sponsoring the projects, are often demoted from a number of important d~ cisions. The paper describes the concepts and the methods that lay beyond the develoE ment of a "home made" general purpose computer system, the SELF system, made operational in Pisa, Italy. The present achievments and the potential impact of the system are described. INTRODUCTION
A UNIFIED APPROACH A number of large scale government projects such as the construction of nuclear power plants, arch and gravity dams, pipelines, high rise building complexes, etc., face severe design problems, due to the size of the project, as well as to the interface needed between different disciplines, such as structural mechanics, fluid flows,thermQ dynamics, etc. The developing country while ultimately paying for the project, is often unable to perform the technical design of its own prQ jects, and resorts to the services of foreign consulting firms. Many decisions, then, are taken thousands of miles away from the site ,with limited feedback from the users of the facilities themselves. The technical gap that is at the base of such a distorted decision process is often the lack of access of the technical community in the developing country to the sophisticated software necessary for the computer analysis of large scale
The problems of mathematical physics are usually divided into four broad families: elliptic, eigenvalue, parabolic and hyperbolic problems. Without any attempt to rigor or to completeness one may typify the different problems as follows: elliptic: field equations (steady state heat flow, seepage in porous media, irrotational flow of inviscid fluids), elastic behaviour of structures and continua under static loads, etc. ; eigenvalue: vibrations, stability of structures, etc.; parabolic: diffusion equations, transient heat transfer, viscous fluid flows, etc. plastic behaviour of structures; hyperbolic: wave equations, elastic behaviour of structures under dynamic loads. In conventional engineering, each problem requires a particular computational technique, thus leading to specialized technical activities, concerned primarily with the
81
mixed parabolic and hyperbolic;
physical aspects of the problem and with limited crosscomunication. The recent revival of var~ational methods associated with the increasing use of large scale computers has set into motion a trend toward using a unified mathematica l model and a general purpose computer system (') ,('0 ). The most important contributions to the variational formulation of differential boundary value problems were given by Mikhlin (6) for elliptic and eigenvalue problems, and by Lions ( 5 ) for time-evolving problems. Quite independently, outstanding engineering schools put into operation powerful computer systems to solve a broad spectrum of civil and aeronautica l engineering problems ('},(9},(" . ) . S ubstantial research work is under progress (4)( 7 ) ( . }( 'O )('2) to achieve a unitary view of the formulation,approximation,and solutio n of the different Kinds of boundary value problems.
A ( t ) u ( t ) + B ( t ) u ' ( t ) + C(t }u" = f ( t ) In
n n
in
eigenvalue;
Au
AU
in
parabolic;
A (t ) u (t ) +u ' (t) = f( t} ; n , O
where
u"
in
o
a
2
n n
u
2
C (t) linear operators
a(u,n} = (Au,n)
n £V
The b ilin ear form is associated with the inner product,after the inte gration by parts is performed v} a linear functional is introduced, L(v} defined on V; vi}in problem (3) both sides of the equa tions are integrated between 0 and T, after performing the inner product.
(1 )
Problems (1},(2) and (3) are finally placed in the form see ( 4 ) , ( ~ ) , ( 6 ) ;
(2 )
a(u,.p} = L( cp}
(3 )
0
where
u'
in
o
O
Here the depende nce of the solution u from the "space" variables is implicit, while the dependence from the "time" parameter t is explicit. Similarly, the need for the solution u to satisfy " space " boundary conditions is cumulated in the r equirement for u to belong to the subspace D(A} of H whil e the initial conditions are explici~ 'I~e differential problems (1},(2},(3) are transformed with the following operations; i} both sides of the equations are innerly multiplied by a "test" function n belonging to HA; ii}the inner product (Au ,n), implying an in tegration ove r derivatives, undergoes an integration by parts, that lowers the or der of the derivatives a n d introduces the boundary conditions; iii}a subset V of the function space HA is taken into account, where the functions have derivatives square summable up to a certain order ; that is referred to as a Sobolev space , and is proved to be den se in HA ( 6 ) ; iv}a bilinear form is introduced, a(u ,n} de fined in VxV, such that
HA the completion of the subspace D(A) with respect to the energy norm;. In the given n otat ion, the above mentioned problems are represented in the following compact form ; f
u(o}
B( t},
In the following, the problems so far presented will be introduced in the concise functional analysis notation. For sake of brevity, we do not carry all the prelimin~ ry mathematica l notation, leaving to the r~ ferences the precise definition of the individual problems. Let A be a linear differentia l operator , symmetric and of the elliptic type; H a Hilbert space of functions; D(A} the subspace of functions of H opera ted upon by A; U,v functions of D(A}, defined on an open , bounded subset n of the Euclidean space RN; (u,v) the inner product of u and v in H ; Ilull (u,v)~ the norm of u in H; [u,v] = (Au,vl the "energy" inner product IlulIA= [U,U]2 the energy norm of U;
Au
u
at
A UNIFIED MATHEMATICAL MODEL
elliptic ;
u(O}
n
(4 )
where ~ is a properly defined subset of V . The conditions for the existence and the uniqueness of the solution of problem (5) aregivenby(5} ( 6).
au at
82
The differential problem (4), on the other hand, is transformed by Galerkin's method. The function u(t) 1S assumed to be of the separable form u(t) = w(x)y(t)
x£R
is organized and structured in three different areas: 1. Partition of the procedure into several levels of operations; 2. assembly of the subroutines into modular blocks; 3. structuring and processing of data sets. A brief outline follows of the main features of the SELF system. Partition of the procedure. One of the most attractive features of syste~ is the capabi lity of solving different problems. In the overall procedure, the operations that are specific to each problem are inserted "in parallel" into the flow of operations that are common to all problem. The macrof l owchart for dynamic analysis(eq. (9)) is shown in fig.l. Each block there ig dicates a phase of the procedure that oper~ tes independently from the other ones. A phase is in turn partitioned into functions. Functions can be of a general kind, or specific to a problem. Finally, the functions are broken into a string of elementary matrix operations. Modular blocks. The subroutines are set up and collected into modules so to minimize the number of subroutines and to adjust the length of the module to the primary memory available. In order to reduce the number of subroutines, the elementary operations are collected in a ~ tility module that can be accessed from each phase and function. A number of subroutines are sufficently general to be usedin more than one function. The modules are dynamically a llocat ed into primary memory from the monitor program, a£ cording to the flow of the procedures. The modules over lap to each other. Within the phases, the general functions and the functions specific to the problem are collected into different modules; thus for each problems only the related modules are allocated. Data structure. The working data , on which is performed the bUlk of operations , are parametrized by the number of nodal variables, that changes with the type of problem and eventually from node to n ode. Thee major phases have a monitor program : that, anticipating the operations to be performed, shifts into primary memory the data to be processed. The purpose is to minimize the number of accesses to the slow secondary m~ mory: thus the data are retrieved in blocks that are as large as required for the completion of the operation, or, otherwise, as large as the primary memory available .
N
performing the same operations, i) to v) as before one obtains, using an analogous nota tion a(t;w,
=
L (t; cp)
(6 )
APPROXIMATION AND SOLUTION Problem (5) is approximate by a finite dimensional problem where the functions u and n are sUbstituted by piecewise polynomial functions. The procedure is known as the finite element method( " ). For the computational details, and for an appraisal of the er.ror bounds see ( 3 ) , ( 8 ) . Calling ~ the vector of values of u at a fi nite number of points, and ~, ~ the vectorn of first and second derivatives with respect to time , the approximating problem takes the following f orm, in matrix notation elliptic,parabolic
Mu
P
(7 )
eigenvalue
Mu
AU
(8 )
mixed parabolic and hyperbolic M~(t) +N~(t)+Q~(t)=P(t) By applying the Fourier transform to eq.(9) one obtains: (10) where u(w) is the Fourier transform of ~(t), p(w) i; the Fourier transform of p(t), and Q(w) is a vector of terms that take into ac count the initial and the final conditions on the t i me v a r i ab 1 e. E q . (1 0) to sol v e d for a discrete set of values of the frequency w. Applying to u(w) the inverse Fourier transform one finally obtains u(t). Direct and inverse (fast) Fourier tra~sforms are comp!:!. ted by the Cooley-Tukey algorithm.
A MODULAR COMPUTER SYSTEM The flow of operations in the SELF system
83
INPUT
'--71> -:.,
no
~~~2:s;~2J~~~tt!;l~~~~~~bf >
" '\
~l-'.~t--t'-~~+~.~ +-:.-1._. --r~r-----;'~.~k--------'i<:-:-i-;f:-k'i
for every Wj
f-
assembly of the matrix
M+iw.N-w 2 Q J
i
J
L
Set up boundary
conditions for eq.(ll) Fi g. 2 - Shielding structure oC 8. nuclear reactor core.
Solution of eq.(ll)
sto rage into file of the solution u(w.) -
J
for every loading
,
condition P.(t)
,
retrieve P.(t) from file
Fourier transform
,
T(P.(t))
,-
Fig.
Fig. 1 - Macro flowchart for dynamic structural analysis .
84
3 - Enclosing shell of a nuclear reactor.
APPLICATIONS
a compact 3000 ca rd package. With respect to similar, large scale finite element sy stems like NASTRAN , ASKA, STRUDL,the ratio between numbers of cards ranges between VIO to 1/ 30, while the ratio of analytic capabi lities goes from 0 . 6 to 1. The sUbstantial economy in the number of cards is due to s~ veral factors; in order of relevance: i) the SELF system was designed for modern large scale computers, with time-sharing modes of operation and hardware based "virtual" memory; the handling of data in and ut of . core is reduced to a minimum; ii)the SELF system, solves internally, a small number of general mathematica l pr2blems; the variety of physical problems is confined to the input and output of data; iii)the continuity of the team work ; it l S proved nowaday that changes in the team composition or in the supporting contract affect the developing program with uncon trolled mushrooming.
The SELF system was developed for research purpose, and in such form released for general use. The modular structure, however, is such that many problems of practical interest can be solved by introducing minor modifications or extensions to the bulk of the system. Some institutions,then, have adopted the policy of extending the SELF s~ stem by their own means to fit their problems and their computational facilities. Here are some applications collected from various users. In fig.2 the shielding struc ture of the core of a nuclear reactor is shown, in its section view, together with the mesh size used for the analysis. The temperature distribution in the body was computed first; then a thermoelastic analysis was performed on part of the structure. In fig.3 is the perspective view of the outer shell of a nuclear plant. The s hell is stiffened by a large box girder ring(no shown here),and was analyzed for stresses and deflections under several loading con ditions. Fig.4 shows the plan view of the roof of a new cathedral ana lysed for stresses under wind loads. The 150' diameter reinforced concrete roof is corrugated, stiffened by curved panels, and carries a 30' high slab. Fig.5 shows me fluid section of a plasma gun. Assuming ste~ dy flow, the velocity distribution in the fluid was computed using the fine partition in triangles shown in the figure. The collection of those few examples indic~ tes, beyond the details of the individual problems, the versatility of a computer system, such as the SELF s ystem, as a broad range problem solver.
Another relevant characteristic of the SELF system is its open box availability.The sol ving technique can be learned, and improve~ not just borrowed. The impact of such policy on education and applied re sea rch i s obvious . Public institutions, that often carry the burden of che cking and controlling complex projects, have now free access t o a relia ble "home made" computer packa g e. The conclusion derived here, after the dev~ lopment of the SELF system, are: i) a great importance is attached to the planning phase of the development:the s ~ lection of the mathematical model, the structure of the syste m, the direction ci the possible extensions, etc.; ii)the development should be phased in con di tions of high efficency: a compact teeJll, concentrated in one place, a modern, la~ ge scale computers, broad availability of computer time, continuity of the resear& effort.
EVALUATION AND CONCLUSIONS The development of the core of the SELF sy stem took two years of a five man team. A sUbstantial amount of initial planning highlighted the basic functions of the sistem and the interface with possible extensions of both the analytic capabilities of the pr2gram and more sophisticated input output ~ rations. Thus, any modification the individual user may want to introduce can be addro as a branch to the core of the system wi thout upsetting the structure of the program. The ind i vidual subroutines are optimi zed with respect to number of instructions and core occupancy; the programs are written in FORTRAN IV for simplicity of transfer from one computer to an o ther. The system is
When two such conditions are met, a relative ly small funding effort, combined with an appropriate release policy that favors'~all outs" of the basic research toward applied research, may produce very rewarding resuhs. The developing countries are often mislead, by the so called technological gap ,i nto tr~ ing to upgrate their entire technical comm~ nity. Such policy often leads to dispersal of energies and to discouragment in front of the fast paced advances in technology. The concentration of efforts, on the other hand,
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Fig.
y
view of the
4 - Plan of a cathedral . roof
[cm]
18
O~-----------------------~==~~~~402±~~~to50--x~[cmJ o
. . to elements . . 5 - Plasma gun: partit10n 1n flg
86
is certainly a short term solution; but it may lead to reducing the gap in a number of important areas, with precious savings for the country. Time is thus bought for the fairly slow diffusion process of the knowhow, and confidence is gained in the possibility of producing original contributions.
(5) Lions J.L., "Equations differentielles operationelles et Problemes aux Limites' Springer, Berlin, (1961). (6) Mikhlin S.G. ,"Mathematical Physics, an Advanced Course". North Holland, Am s t er d am, (1970).
AC KNOWLED GI4ENT
(7) Strang G., Fix G., "A Fourier analysis of the finite element method~ To appear in the Prentice-Hall Mathematical series.
The present research was partly supported by grant n055 0 from NATO Scientific Affairs Division.
(8) Strang G., "Approximation in the finite element method". Num. Math., 19,2, (1972~ 8l-98.
REFERENCES (1) Argyris, J.H., "The impact of the digital c o mputer on the engineering sciences". Part I', Aero. J.Roy.Aero.Soc., Vol. 74,pp.13-41, 1970.
(9) Strudl I., "Internal logic manual". IBM Publ. n.360-D 16.2.015, White Plains, N.Y., (1968).
(2) Carmignani C., Cella A., Lami V., Bozzi R., "11 sistema SELF/2 per il me todo degli elementi finiti". I.E.I. Re port B72-6, (1972).
(lO)Whiteman J.R., Barnhill R.E., "Finite element methods for elliptic mixed bOu£ dary value problems containing singularities". Proceeding EQUADIFF 3, Brno, Czechoslovakia, (Au~.1972).
(3) Cella A., "Approximation techniques in the finite element method". To appear in the CISM Springer Series, Wien.
(ll)Zienkiewicz, "The finite element method in engineering sciences". Mc Graw-Hill, New York, (1971).
(4) Cella A., Morandi Cecchi M., "An extended theory for the finite element method". Proceedings of the InternatiQ. nal Conference on Variational Methods in Engineering. Southampton (Sept .1972)
(12)Zlamal M., "Some recent advances in the mathematics of finite elements". Int. Conf. on Mathematics of Finite Elements, Brunel University (April 1972).
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