The superposition theory is adopted to study the boundary value problem of needle domains in barium titanate single crystals, which decomposes into an infinite-medium solution and a complementary solution. The model of an equivalent edge dislocation and line charge is adopted to calculate the fields due to the discrete needle domains in an infinite medium. The complementary solutions are obtained from a linear elastic boundary value problem using the finite element method. The influence of boundaries on the fields and arrangement of the needles in thin barium titanate single crystals is analysed, by considering traction-free boundaries and using the plane stress approximation. According to the principle of minimum potential energy, it is found that pairs of parallel needle domains can be stabilized if the effective charge associated with the needle tips is reduced to 48% of the material’s full polarization charge. The evolution of comb-like arrays of needle domains under compressive mechanical load is simulated, and the results are found to be qualitatively similar to experimental observations which have not previously been explained theoretically.

采用叠加理论研究了钛酸钡单晶中针状区域的边值问题，该问题分解为无穷介质解和互补解。采用等效边缘错位和线电荷的模型来计算无限介质中由于离散针域引起的场。互补解是使用有限元方法从线性弹性边界值问题获得的。通过考虑无牵引力边界并使用平面应力近似，分析了边界对薄钛酸钡单晶中针的场和排列的影响。根据最小势能原理，发现如果将与针尖相关的有效电荷减少到材料的全极化电荷的48％，则可以使成对的平行针畴稳定。模拟了针状域在压缩机械载荷下的梳状阵列的演化，发现该结果在质量上与以前没有理论解释的实验观察结果相似。