Development of robust numerical methods
as applied to welding processes
V.V. Yakovlev, I.V. Pershin, S.M. Shanchurov, G.I. Shishkin
United Metallurgical Company, Ural Branch Russian Academy of Science, Ural State Technical University, Ekaterinburg, Russia
Abstract. Processes of welding and surfacing are characterized by presence of the powerful concentrated sources of heating because of what there are significant temperature gradients, especially near to sources of heat. For the solution of such problems application of classical both analytical, and numerical methods gives big uncontrollable error. It is necessary to develop and apply robust methods, i.e. special methods which do not depend on features of concrete physical process, and include these features. It allows to get the solution and its derivatives (thermal fields and streams of heat) with preset accuracy.
Welding and surfacing are technological processes which are characterized by presence of powerful thermal fields, streams of heat and substance, thus simultaneously there are firm and liquid phases, there are complex hydrodynamic flows, various chemical reactions proceed. The complicated geometry of products, heterogeneity of welded and surfaced details, presence of small inclusions (pores) play big role. Heating processes play important role during welding and surfacing which define presence of all other processes.
Actual condition of the further improvement and wider application of processes of welding and surfacing is not only continuation of development of theoretical bases of welding in using of advanced achievements in various areas applied and fundamental sciences, but also development of highly effective methods of modeling of welding processes. This problem becomes especially important in connection with rough development of computer engineering and computing technologies. The modern computer has such opportunities as speed, memory, the standard software, a set of the specialized software programs and peripheral devices, which allow using it in a mode of real time for solution of wide range of theoretical and practical problems of welding.
In the modern welding literature practically there are no the works devoted to methodology of application of computing experiment which is enough developed in detail in mathematical physics and calculus mathematics. A.Samarsky (1) has brought the big contribution in development of methods of computing experiment. There was a necessity to generalize available results, to concretize approaches and methods of use of modern computer equipments and computer technologies of modelling with reference to research of problems of welding on the basis of the accumulated experience (2).
Special difficulty in research of the majority of technological processes of welding and surfacing is represented with presence of singularity: the investigated physical quantity sharply can change the value in a small spatial and time interval. In thermal processes such quantities are the temperature and streams of heat, which values are sharply varied near to a source of heat or border of body. In diffusion processes the value of concentration of substance strongly can vary on border of two mediums. Speed of movement of molten metal in a pool are considerably changed near border of a pool in research of hydro dynamical processes. Similar physical effects necessary to study especially carefully as they are directly connected to occurrence of defects - cracks, pores, residual pressure and deformations, and also other defects in welding and surfacing. These processes are referred to as processes with boundary layers if a similar sort of effects arise on borders of investigated areas owing to the appropriate boundary conditions; or with internal transitive layers if effects arise inside area - for example, in use of multi-component materials with various thermo-physical characteristics (3).
Nowadays a great number of physical and mathematical models for determination of temperature fields and streams of heat in various processes of welding and surfacing were developed. For solution of appropriate systems of differential equations a lot of various analytical and finite-difference methods of the solution were developed, on the basis of which a plenty of applied programs for computers were created with the purpose of realization of various calculations and numerical experiments. Thus by virtue of such mass development and popularity of computer technologies, illusion have received a wide circulation, that if the problem was solved on a computer with using of standard programs the received result fully complies with the validity.
It is necessary to show, that the majority of modern technological problems of welding and surfacing can not be solved with required accuracy by using classical analytical and finite-difference methods which are applied in the majority of computer programs. Application of classical analytical and numerical methods at modeling processes of welding and surfacing gives the big uncontrollable error of the solution and especially its derivatives which are commensurable on value, and quite often considerably surpass the required solution. Thus it is impossible to determinate beforehand, at what values of parameters of process there will be a sharp increase of size of an error. For the big circle of problems, determination of derivatives (streams of heat and substance) with accuracy necessary for practice has decisive importance. Speed of cooling during welding is a determinative factor of final structures and properties of welds. Thus performance of some conditions is necessary: the difference of temperatures on various sections should not exceed technologically necessary, the temperature should not be more specified, stay of a material at the given temperature during necessary time etc. Realization of numerical calculations with accuracy below required will not allow to receive the results appropriate to conditions of practical use (4).
Very actual and difficult is a problem of an estimation of accuracy of the received approximated solution at application of numerical methods for the solution of the differential equations which are included in mathematical model of technological process of welding. The actuality of this problem consists of the received approximated solution can differ from true solution in 2-3 times, and for some problems - in tens thousand times. Difficulty of a problem is what to receive the obvious analytical solution of such problem even in case of essential simplification of physical and mathematical models of process of welding frequently it is not possible; precisely as realization of nature experiments with investigated object is quite often impossible. Hence, to compare the approximated numerical solution to any other independent result is extremely difficultly.
Accuracy of the approximated solution of a problem at using of classical finite-difference scheme depends on the chosen numerical method, parameters of a differential problem and parameters of a numerical method. Parameters of a differential problem are coefficients of the differential equations which are included in mathematical model of process of welding, parameters of a numerical method - number of nodes or a step of finite-difference mesh. Cases when accuracy of the received approximated solution does not depend on number of nodes finite-difference mesh, but only from parameters of an initial physical problem are frequent. In other words, the error of the approximated solution does not decrease with increase of number of nodes of finite-difference mesh. Classical finite-difference schemes have the limited area of using; the sizes and borders of this area depend on a ratio of parameters of the differential problem, the chosen numerical method and required accuracy of the solution. Outside of limits of this area classical finite-difference schemes do not give necessary accuracy, and application of special methods is necessary. For problems of welding and surfacing such cases are enough typical (5), (6).
Similar problems arise at use of classical analytical methods for research of behavior of thermal fields and streams of heat nearby from a source of heating.
During the past few years robust methods are developed, i.e. special methods which do not depend on features of concrete physical process of welding and surfacing, not only are steady against change of initial parameters of a problem and various features of behavior of the solution, but also include these features (7). Such robust finite-difference and analytical methods allow to solve problems of welding and surfacing with a necessary and preset accuracy.
As mention above, the majority of processes of welding, surfacing, and also many others thermal or diffusion problems at development of physical and mathematical models result in a class of differential tasks with the transitive or boundary layers, named also the singularly perturbed problems. The most frequently similar problems by transformation of differential equation, which entered to the mathematical model, are reduced to the tasks, containing small parameter at the senior derivatives, which can accept any values, including, aspire to zero (5).
For example, we shall consider the problem which has arisen by development of technological process of cladding rollers. In this case process of heat exchange is a process of heat exchange in the cylinder with a mobile source of heat on border. Such problem we shall name a problem about heating the cylinder by the concentrated normal - circular source of heat moving on a screw line.
The process of heat conductivity in cylindrical body with moving heat source is described by the three-dimensional equations of heat conductivity accordingly in cylindrical system of coordinates:
We designate: T is an unknown temperature, a is a coefficient of temperature conductivity, l is heat conductivity, R is a cylinder radius, Z is a length of a cylinder, j is an angular coordinate, t is time. Function y() characterizes the power of a moving heat source and a rule of its distribution. The sign “~” is defined that the variables are measuring. We will use the non-measuring variables and we introduce the characteristic parameters – characteristic time of process t and characteristic length L. The characteristic length L is defined as follows:
, where is the area of a surface of a moving heat source, and S is the area of a surface of cylinder. We receive the equation in the non-measuring form:
where is Fourier criteria.
It is necessary to note, that characteristic size L can vary in a wide range from 10 up to , as the length of the cylinder can vary from 0.1 m up to 2 m diameter of the cylinder can vary from 0.1 m up to 1 m, the size of a source of heat from 1 mm up to 100 mm, duration of process varies from several seconds till several hours. In the given problem Fourier criteria can have various values in a range from 100 up. In such problems of welding and surfacing the physical sense of parameter e at the senior derivatives corresponds to Fourier criteria, or a value reversing to Peclet criteria. The problems arising at using a numerical method of the solution of these problems are connected to that; this parameter can accept any values in a wide range of sizes from up to.
The main difficulties, which are connected to numerical solution of the differential equations with parameter at the senior derivatives, we shall look after on the one-dimensional stationary equation of heat conductivity with moving heat source:
,
which have an obvious solution: , which was used for comparison with the received approximated numerical solution.
In a case, when the parameter is commensurable or is much greater to 1, the solution of such problems does not make difficulty; they are well solved by standard classical finite-difference methods with good accuracy. Behavior of the obvious solution and errors of the approximated numerical solution at various values of parameter e received with use classical schemes at values are submitted in Figs. 1 and 2:
Fig. 1. Fig. 2.
In cases when the parameter is small or aspires to zero, the numerical solution differs from the obvious solution on finite quantity, and its derivatives aspire to infinity, an error of the approximated solution and its derivatives in this case look as follows Figs. 3 and 4:
Fig. 3. Fig. 4.
Processes of welding and surfacing include all spectrums of such problems.
There is developed the technique, which allow determinate the accuracy of the numerical solution and its derivatives depending on finite-difference methods and initial parameters of a physical problems of welding and surfacing.
It is shown, that classical standard finite-difference methods do not allow receiving the approximated solution and its derivatives with the given accuracy, and thus errors can uncontrollable increase. Areas of applicability classical finite-difference methods are determined and investigated depending on parameters of a numerical method and parameters of a physical problem. The behavior of errors of the approximated solution and its derivatives, both for classical, and for special numerical schemes is investigated. Schematically these areas for classical difference schemes are submitted on Fig. 5:
Fig. 5.
Here: the zone 1 – a zone of regular solutions of an applied problem, a zone 3 – a zone of singular solutions, a line 2 – the border of their separating. It is obvious, that it not a precise line, but the smeared area. The zone 4 is the area in which the width of a boundary layer is much less than the step difference grid, but the solution in it is too singularly.
These investigations allow solving two problems:
Problem I. At the chosen numerical method of the solution of system of the differential equations and the given parameters of this method (number of nodes of finite-difference mesh) to specify those values (area of change) parameters of initial physical process at which the approximated solution will be designed with the given accuracy;