It is well known that the weld bead width becomes wider and the weld pool hangs down as the circumferential welding of small diameter pipes progresses, if the constant welding conditions are maintained over the full joint length and if the compressed backing gas is not supplied. In order to obtain a uniform and no hang-down weld bead over the entire circumference of the pipe, the welding conditions such as the welding current, welding velocity and backing gas pressure should be adjusted and optimized as the welding proceeds. One of the important problems to be solved in the welding engineering is to develop the mathematical method for the determination of optimum process parameters. Thus a mathematical modeling of the welding process for determining the temperature distribution in the weld metal and the surface profile of the resultant weld is indispensable to optimize the process.
First, in order to estimate the optimal process parameters such as the welding current and welding velocity in circumferential GTA welding of thin pipes, the objective function was chosen to maintain a uniform bead width over the full circumferential joint, while the constraints consist of the capacity limit of the welding power source and equipment used.
The linear complementary problem(LCP) with Lemke's pivoting algorithm and Powell's unconstrained search method with the sequential unconstrained minimization technique (SUMT) was applied to evaluate the optimal welding current with a given welding velocity and the optimal welding velocity with a given welding current for a required bead width respectively. Here, the analytical solution of heat conduction equation with a Gaussian heat source was derived and applied to calculate the temperature filed and consequently to determine the weld width in circumferential welding of pipe workpieces.
An efficient parameter optimization model for the numerical heat conduction flow was proposed to evaluate the optimal welding current with a given welding velocity for a required bead width. Its solution was obtained by employing the steepest descent method(SDM), where the initial value of the welding currents in the middle part of the pipe were interpolated by the least-square approximation method of the second order. Here, a transient three-dimensional finite difference model(FDM) of the heat conduction flow was developed and effectively used for more accurately calculating the temperature field in the pipe workpiece considering the temperature-dependent thermal properties of the workpiece and consequently for determining the resultant bead width in circumferential welding of the pipe workpiece. In order to minimize the computing time required for solving the FDM equations as much as possible, the alternating direction implicit(ADI) scheme which makes use of the tridiagonal matrix algorithm(TDMA) was adopted. Based on the characteristics of the pipe welding process, the periodic boundary condition was applied to calculate the temperature distribution in the $\theta$ direction, For treating the moving heat source, the grid meshes with variable spacings were regenerated at each time step. In order to decrease the interpolating error by grid remeshing, the temperature values at new meshes were interpolated from those at old meshes by using the periodic spline function. The temperature-dependent thermal properties, the latent heat and the convective and radiative boundary conditions were considered for calculating the temperature distribution of the pipe workpiece in this model. The calculated sizes of the fusion and heat-affected zone were compared with the observed values after experiments.
The experimental resultant weld beads for three model mentioned above showed that the developed mathematical optimization model can be effectively applied for determining the optimal welding conditions such as the welding current and velocity in order to obtain a uniform weld bead in circumferential welding of thin pipes with a small diameter.
The effects of the gravity on the weld pool can be balanced by the arc pressure and backing gas pressure, and the bead shape and penetration can be completely controlled by adjusting the backing gas pressure. Thus the optimum backing gas pressure is required to prevent the weld pool of pipe workpiece from the burn-through and too much hang-down.
At the first place, the parameter optimization model for the numerical heat conduction flow described above was also used for obtaining the optimal welding current for a given welding velocity which gives a uniform weld bead over the entire circumferential pipe joint. Based on the golden section search method, the developed mathematical optimization model was applied to determine the optimal backing gas pressure for obtaining no hang-down of the full penetration weld in GTA welding of thin pipes. Here, a governing equation of the weld surface profile was derived and the finite difference method with the ADI scheme was employed for calculating the hang-down of the resultant weld.
The experimental results of the weld bead profile showed that the developed mathematical model can be effectively applied for determining the optimal welding conditions such as the welding current, welding velocity and backing gas pressure in order to obtain not only a uniform weld bead over the entire circumferential pipe joint but also no hang-down of the resultant weld in circumferential welding of thin pipes with a small diameter and high thermal conductivity.
용접공정에서 중요한 문제중의 하나는 실험에 의존하지 않고 최적의 용접공정 변수를 결정할수 있는 수학적 모델을 개발하는 것으로서, 일반적으로 직경이 작고 열전도성이 좋은 파이프를 전 원주에 대하여 일정한 용접전류 및 용접속도로서 원주용접을 하면 용접이 진행됨에 따라서 용접비드의 폭이 넓어지고 또한 백킹가스를 공급하지 않으면 용융부가 아래로 처지므로 용접부의 질이 저하된다. 따라서, 균일한 비드폭와 처짐 없는 건전한 용접부를 얻기 위하여는 용접전류, 용접속도 및 백킹가스 압력등의 용접변수를 최적화하고 제어해야 할 필요성이 있다.
이러한 문제를 해결하기 위하여 본 연구에서는, 우선 용접부의 온도분포를 간단하게 결정하여 용접비드폭을 결정하기 위하여 Gaussian 분포의 열원을 갖는 3차원 비정상 열전도 방정식의 해석해를 파이프의 주기성 조건을 근거로 하여 유도하였다. 전원주에 대하여 균일한 용접비드 폭을 얻기 위하여, Lemke의 피봇팅 알고리즘을 사용한 선형보상문제는 주어진 용접속도에 대하여 최적의 용접전류를 결정하는데 사용되었고, 또한 SUMT를 적용한 Powell의 비구속탐색법은 주어진 용접전류에 대하여 최적의 용접속도를 결정하는 비선형문제에 유용하게 사용 되었다. 여기서 상기 두 방법은 매우 계산시간이 적게 소요되는 장점은 있으나 온도분포를 구할때 온도에 따른 물성치 변화를 고려하지 않았다.
그리하여 온도에 따른 물성치 변화 및 잠열을 고려한 3차원 비정상 열전도 방정식을 ADI 차분법을 사용하여 파이프의 온도분포를 결정하여 용접비드 폭을 결정하였으며, 여기에서는 파이프의 주기성 조건의 적용으로 인하여 계산시간이 많이 절약된다. 전원주에 대하여 균일한 용접비드 폭을 얻기 위하여 급강하법이 주어진 용접속도에 대하여 최적의 용접전류를 결정하는데 사용되었고, 계산시간을 줄이기 위하여 선형보상 문제에 의하여 결정된 용접전류를 최초의 데이터로서 사용하였다.
황금분활법에 의하여 개발된 수학적 모델은 용접 용융부의 처짐이 없도록 최적의 백킹가스 압력을 얻는데 적용되었으며, 용접부 표면 형상에 대한 지배방정식을 변분법에 의하여 유도하였으며 그것은 ADI 차분법에 의하여 해를 구하였다.
결론적으로 용접비드 형상에 대한 실험결과들을 볼때, 본 연구에서 개발된 최적화에 관한 수학적 모델은 직경이 작고 열전도성이 좋은 파이프를 원주용접할때 전 원주에 대하여 균일한 용접비드 폭과 처짐이 없는 용접부를 얻기 위하여는 용접전류, 용접속도 및 백킹가스 압력등의 용접조건을 결정하는데 유용하게 적용 될수 있었다.
