COMPUTER-ASSISTED SIMULATION ENVIRONMENT FOR PARTIAL- DIFFERENTIAL-EQUATION PROBLEM: 1. Data Structure and Steering of Problem Solving Process

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1 Transactions of JSCES, Paper No COMPUTER-ASSISTED SIMULATION ENVIRONMENT FOR PARTIAL- DIFFERENTIAL-EQUATION PROBLEM: 1. Data Structure and Steering of Problem Solving Process Choompol BOONMEE and Shigeo KAWATA Dept. of Electrical Eng., Nagaoka University of Technology (Nagaoka , Japan) Abstract In this paper we present a new data structure which gives a steering capability to a computer-assisted scientific- simulation environment or a problem solving environment (PSE) for partial- differential- equations (PDEs) - based problems. One of the important key issues in PSE researches is a steering capability including the interaction between a PSE and a user. In order to tackle this key issue, the new data structure is proposed to introduce the process steering capability to the PSE. The data structure is used to describe PDEs, boundary conditions, the initial conditions, all processes including PDE discretization, equation manipulation and program generation, and a generated program flow. In the data structure each PDE, each term, each symbol, each process and each component of the generated program flow have their meanings, which are linked and referred by pointers. We apply the data structure to a PDEs PSE called NCAS system. Based on the proposed data structure, the steering capability is realized, and the PSE communicates with the user smoothly and flexibly through an interactive visual interface. Key Words: Data structure, Problem solving environment, Steering, Numerical simulation, Program generation, Process visualization 1. INTRODUCTION A problem solving environment (PSE) can be considered to be a computer software system that provides a computational facility to solve a target class of problems[1]. One of the important key issues in PSE researches is a steering capability including the interaction between a PSE and a user[1,2]. The facility should communicate with users smoothly and flexibly. In the past, several PSEs have been studied and developed[1-8]. Most PSEs do not support a process steering capability, which allows users to steer the problem solving process. In this paper we focus on a PSE for partial- differential- equations (PDEs) based problems. A number of well-known discretization methods and computational techniques have been developed to solve PDEs. When a user solves a problem, the user can and should choose one computational method which matches the problem, and may change the problem solving method or process. This process change or steering requirement addresses PSE researchers to work on a PSE which has a steering capability. In this paper, we present a new data structure to express or to describe PDEs-based problems and problem solving processes: the new data structure is used to describe PDEs, boundary conditions, the initial conditions, all processes including PDE discretization, equation manipulation and program generation, and a generated program flow. In the data structure each PDE, each term, each symbol, each process and each component of the generated program flow have their meanings. The meanings are linked and referred by pointers. Based on the proposed data structure, the PSE provides users the steering capability, and communicates with the user smoothly and flexibly. Received September 4, 1997; in revision December 22, 1997;published January 12, 1998 In the conventional object-oriented paradigm, for example, a 'class' is a kind of template to define a behavior of a particular type of object. We may use a 'class' to define symbols' definitions, for example. However in the conventional objectoriented programming languages (OOPL) (C++, Java and so on) the 'class' definition is static and considered as a fixed global one. Once a 'class' is defined, the definition can not be subsequently changed during the job execution. On the other hand, data structures employed in the previous PSEs[3,6,7] have the similar feature to that of the 'class'-based data structure. When we employ the conventional data structure, the cycle of the re-editing of source file and the compilation must be repeated. For the steering capability the 'class' definition should be also changed dynamically. For example, a symbol definition or a discretization method or a part of computation process may be changed during the problem solving process; a symbol may appear many times in PDEs used. When its definition is changed by a user, the 'class' definition itself should be changed. This requirement can not be realized by the conventional data structure, and leads us to construct and use the new data structure which allows users to change the 'class' definition itself. The data structure proposed in this paper is a tree-type structure. In addition to realizing the steering capability to PSEs by the data structure, an appropriate grouping of terms or PDEs or processes or computation programs generated is easily realized in our PSE. The grouping is important for a structured-program generation and for a compact effective visualization in PSEs. In addition, input or modification errors of a PDEs problem in a PSE can be avoided by the data structure, because each data has its meaning.

2 We have developed a PSE of a computer- assisted- numericalsimulation environment called NCAS system[2,4], in order to analyze PDEs-based problems and support scientists and engineers in scientific computation. In NCAS system the visualization and steering capabilities are newly introduced and realized to support users who work on PDEs-based problems. We present these capabilities in detail in ref. [2]. In order to realize especially the steering capability, the new data structure is required and proposed in this paper. We discussed the data structure in the following sections in detail. NCAS system inputs a problem from users through its graphical user interface. The users start their jobs from the definitions of symbols used in their problems which may include dependent variables (physical quantities), parameters, computation step number, computation domain and boundaries, the initial conditions, and so on. Then NCAS system initially chooses a standard problem solving process and method for a given PDEs-based problem automatically. The system visualizes all the processes, including problem description, PDEs dicretization and manipulation processes and program design process, and also program flow. NCAS also provides a steering capability to the users. The users can verify and can steer the problem solving process. Lastly the system generates a C language program to perform the computation. 2. DATA STRUCTURE AND STEERING We propose a proper tree-type data structure which consists of two types of components: symbol definition and symbol instance. The symbol definition can be considered as a classlike object in which all properties must be declared. The symbol instance has a pointer to the 'symbol definition' and can be considered as an instance of the 'symbol definition', similar to but different from an instance of a class object in the object oriented paradigm. The both components exist in the tree structure and belong to the same root node. These features are illustrated in Fig.1. In the conventional OOPL the generation and the initialization of a new instance are performed by 'new' operator and the constructor. Once a new instance is generated, it has all properties declared in its class object and behaves independently from its class object. The 'class' definition is static and considered as a fixed global one. Once a 'class' is defined, the definition can not be subsequently changed. In our proposed data structure both the symbol definition and the symbol instance exist in the tree structure and the relation between symbol definition object and symbol instance object is accomplished via a pointer-link mechanism. Instance objects have pointers (the circles inside component boxes with an arrow line in Fig.1) which point to their class objects. When the user steers or modifies a symbol definition for example, the definition itself is modified and at the same time all instances linked to the definition are changed by the pointer-link mechanism. This and the dynamical modification of the symbol definition can not be realized by the conventional data structures described above. Fig.1 presents some symbol definitions on the left part ('x','u' and 'x-velo') and some symbol instances on the right under the root node. In NCAS system the symbol definitions and their symbol instances are always linked by the pointers. All symbol instances have the pointers which point to their symbol definitions and leave all jobs to their symbol definitions. When a symbol definition is modified by users, the changes propagate to all of its instances at the same time. Fig.1 An example tree-type data structure for the symbol definition and the symbol instance proposed in this paper. The mechanisms of inheritance are also achieved by these pointers. In Fig.1 the symbol 'u' is an inheritance of the symbol 'x-velo'. In our PSE, we use the data structure in presenting PDEs, including boundary conditions and the initial conditions. In order to express the difference term in Fig.2, the components have the pointers to their definitions. For example, in Fig.2 the component T has a pointer to its definition object, that is the dependent variable 'T' of temperature. In the definition part the information required about the symbol 'T' (e.g. the meaning description in English, the position defined on one mesh, and the solving method: this set is called a Solverset) is included. By establishing and using the reference pointer to each component, the PSE knows or finds immediately the information about the object. This quick response is one of advantages of the data structure proposed in this paper, and makes the flexible and smooth interaction between the PSE and the user. Fig.2 Data structure used in symbols and mathematical expressions.

3 As the data structure shown in Figs. 2 and 3, hierarchical structures are constructed appropriately for mathematical expressions. Fig.3 shows a part of equation which consists of subgroups. The appropriate grouping is realized by the data structure and is important for a structured-program generation in PSEs. Fig.3 Grouping of data structure. In the grid structure information, three subsections are included. They posses symbol definitions referred by symbol instances. The subsection of 'Definition of space variables' has a definition of the independent variables in space. The subsection of 'Definition of boundaries' defines all boundaries used in boundary conditions. Each boundary is specified by its unique name. The boundary name is used for the description of boundary condition as shown in Figs.4 and 5. The subsection of 'Layout figure of grid and boundaries' shows an approximate shape of grid structure in the computation space and the location of boundaries in the computation space. Fig.5 shows an example for the data structure for the boundary condition in NCAS system. The data for the boundary condition also has a pointer to the boundary definition object, in which the position in a space grid is defined. Therefore the boundary condition can be discretized in an appropriate way by using the information. In addition, input or modification errors, which may be made by the user, can be prevented by using the data structure in the PSE. For example, in Fig.3 the user can not substitute n (the time index of T ) with other symbols which cannot be used as a time index. Since the component has a pointer to what it refers, it is easy and immediate to find out the meaning of the symbol and to know what substitution can be allowed or cannot be allowed to the component. It is also easy to implement the engine of discretization and algebraic manipulation, since they can find out the meaning of each component immediately. The description of PDEs problem is also described by the proposed data structure. The input description consists of a grid information (mesh and boundary definitions), a time variable definition and dependent-variable definitions (physic quantity definitions) as shown in Fig.4. An input /output file information and a parameter definition are also included but not displayed in Fig.4. Fig.5 Data structure used in boundary conditions. In NCAS the section of dependent variable definition includes the definition of dependent variables which are solved. Each dependent variable definition has one or more Solver sets. The Solver set means an information set which is required to solve the PDE for the target variable. In the Solver set the computation method for the PDE is specified. The Solver set consists of the following four parts: a target variable, a PDE for the target variable, the region in which the PDE is solved, and the corresponding boundary conditions as shown in Fig.6. Fig.4 Computation description of PDEs problem as an input to the NCAS system. Fig.6 The components of Solver set.

4 In Fig.6 the target variable is T n+1. The PDE and the boundary conditions are solved to obtain the target variable value. The computation is performed for the target variable in the region of 0.0 <x< 1.0, 0.0 <y< 1.0 in this case. Fig. 7 shows on the left the definitions of the space grid information and the time variable information in the PDE problem description, and shows on the right the dependent variable definition of "T info" in NCAS. The "T info." consists of the variable name "T", the meaning of "Temperature", the definition position in one mesh and the Solver set. In Fig.7 the upper suffix of n+1 is added to the symbol T in the second and third terms on the left side of the PDE. In this case an implicit method is employed. Fig.7 Description of a heat conduction problem. Definition of space grid and time variable. An implicit method is employed to discretize the PDE. In NCAS system various processes (discretization process, equation manipulation process, and program generation process) are executed. The processes are visualized and can be steered appropriately based on the proposed data structure. Fig.8 shows how the discretization and equation manipulation processes are presented by using the data structure. The hierarchy of process appears appropriately in the tree structure. The processes 'discretize' and 'arrange' are the instances of the symbol 'ProcD' which is defined for the process that posses other processes as its children. The symbol 'FTCS' is defined for a discretization process, to which applies the forward-intime and center-in-space discretization technique. 'Arrange#1', 'arrange#2', 'arrange#3' and 'Simp' are defined for equation manipulation processe. Fig.8 The proposed data structure is applied to the discretization process of PDEs

5 The NCAS system generates computation programs. The system generates the solver sub-programs for each Solver set and generates the main program which calls these subprograms. The generated program is also presented by using the proposed data structure. Fig.9 shows the solver subprogram 'proc1' and its data structure. Fig.9 Sub program 'proc1' is presented by using the proposed data structure. 3. CONCLUSIONS In order to realize especially the steering capability, a new data structure is required and proposed in this paper. In NCAS system the visualization and steering capabilities are newly introduced and realized to support users who work on PDEsbased problems. The new data structure was employed to express or to describe PDEs-based problems, problem solving processes and a generated program flow including PDEs, boundary conditions, the initial conditions, and processes for PDE discretization, equation manipulation and program generation. In the data structure each PDE, each term, each symbol, each process and each component of the generated program flow have their meanings. The meanings are linked and referred by pointers. Based on the proposed data structure, the steering capability was realized in NCAS, and the PSE communicates with the user smoothly and flexibly. The data structure is a tree-type structure and an appropriate grouping of terms, PDEs and computation programs were easily realized. The grouping is important for a structured-program generation in PSEs. In addition, input or modification errors of a PDEs problem in a PSE can be avoided by the data structure, because each data has its meaning. REFERENCES Gallopoulos,E., Houstis,E., Rice,J.R.: Future Research Directions in Problem Solving Environments for Computational Science, Technical Report CSRD Report No.1259, Report of a workshop on Research Direction in Integrating Numerical Analysis, Symbolic Computer, Computational Geometry, and Artificial Intelligence for Computational Science.,Washington DC., April 11-12, Boonmee,C. and Kawata.S. : Computer-Assisted Simulation Environment for Particle-Differential-Equation Problems: 2. Visualization and Steering of Program Solving Process, Proc. of Conf. on Computational Eng. and Sci., Vol.2, pp , 1997 and also in this issue. Rice, J.R.: Scalable Scientific Software Libraries and Problem Solving Environments, Technical Report CSD TR , Computer Sciences Dept., Purdue Univ., Kawata,S., Iijima,K., Boonmee,C. and Manabe,Y.: Computer-assisted scientific-computation / simulationsoftware-development system. -include a visualization system -, IFIP Transactions A-48, pp , Umetani,Y.: DEQSOL - A numerical Simulation Language for Vector/Parallel Processors, Proc. IFIP TC2/WG 22.5, pp , Cook,G.O., JR: ALPAL, A Program to Generate Physics Simulation Codes From Natural Descriptions, International Journal of Modern Physics C, Vol. 1, No.1, pp.1-51, Baras,P., Blum,J., Paumier,J.C., Witomski,P.: EVE: An Object-Centered Knowledge-Based PDE Solver, In Expert Systems for Scientific Computing, E.N. Houstis, JR. Rice, and R.Vichnevetsky (eds)., Tago,Y.: Expectation to Computational Science (in Japanese), Science of Machine, Vol.49, No.1, pp.78-82, January 1996.