Electrolytic metal coating processes are used to protect products from corrosion, decorative surface finish and other purposes. Electroplated coating has an important quantitative characteristic, which is coating thickness. Since the electric field in the electrolyte is not uniform, the coating thickness at different points on the surface of detail is different. An important task here is to apply a more uniform coating. To solve this problem, it is necessary to calculate the distribution of potentials in a galvanic bath from Laplace's equation. The study aims to increase the convergence rate of the numerical procedure in order to solve Laplace's equation with nonlinear boundary conditions by developing a numerical scheme based on Newton's method. Based on a numerical solution of Laplace's equation with nonlinear boundary conditions describing the potential distribution in a galvanic bath, the thickness of the nickel deposition layer on the surface of a flat metal cathode plate was calculated for different sizes of galvanic baths and anode voltages. A feature of the numerical calculation scheme is the use of Newton's method for the approximate solution of the resulting system of nonlinear algebraic equations with a given accuracy. The obtained results show the effectiveness of the applied numerical method: the quadratic convergence rate of Newton's method gives a time gain of 10 times in comparison with one of the best numerical methods for this type of problem.

中文翻译：

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在电镀液中涂覆扁平金属板时金属分布的建模

电解金属涂层工艺用于保护产品免受腐蚀，装饰性表面光洁度和其他目的。电镀涂层具有重要的定量特性，即涂层厚度。由于电解质中的电场不均匀，因此细节表面上不同点的涂层厚度也不同。此处的重要任务是施加更均匀的涂层。为了解决这个问题，有必要根据拉普拉斯方程计算电镀槽中的电势分布。该研究旨在通过建立基于牛顿法的数值方案来提高数值程序的收敛速度，以解决具有非线性边界条件的拉普拉斯方程。基于拉普拉斯的数值解 如果使用具有非线性边界条件的方程式（描述电镀液中的电势分布），则针对不同尺寸的电镀液和阳极电压，计算出平坦的金属阴极板表面上镍沉积层的厚度。数值计算方案的一个特点是使用牛顿法对具有给定精度的非线性代数方程组的结果系统进行近似解。获得的结果表明了所应用数值方法的有效性：与此类问题的最佳数值方法之一相比，牛顿法的二次收敛速度可提供10倍的时间增益。对于不同尺寸的电镀液和阳极电压，计算了在平坦的金属阴极板表面的镍沉积层的厚度。数值计算方案的一个特点是使用牛顿法对具有给定精度的非线性代数方程组的结果系统进行近似解。获得的结果表明了所应用数值方法的有效性：与此类问题的最佳数值方法之一相比，牛顿法的二次收敛速度可提供10倍的时间增益。对于不同尺寸的电镀液和阳极电压，计算了在平坦的金属阴极板表面的镍沉积层的厚度。数值计算方案的一个特点是使用牛顿法对具有给定精度的非线性代数方程组的结果系统进行近似解。获得的结果表明了所应用数值方法的有效性：与此类问题的最佳数值方法之一相比，牛顿法的二次收敛速度可提供10倍的时间增益。