求解力学中强非线性问题的小波同伦方法拓展研究

The high precision solution of nonlinear differential equations is a key bottleneck problem in the quantitative analysis of mechanical and ocean engineering problems. It is of great scientific significance to conduct research on effective solutions. Wavelet-Homotopy method inherits the strong ability of Homotopy Analysis Method (HAM) to solve strong nonlinear problems and wavelet property of the high approximating precision, which becomes an effective and important numerical method to solve nonlinear problems. However, this method is at the early stage of development only applied in solving large deflection of rectangular plates and steady-state square cavity flow. Wavelet basis is limited to Generalized Orthogonal Coiflets. Combination of wavelet and HAM is limited to the Galerkin method, which the method is limited to solve nonlinear problems subjected to nonhomogeneous linear boundary while calculation domain is limited to regular area, such as rectangular plate and the cavity. The project intends to combine wavelet collocation method and HAM with higher theoretical innovation for the extension research of Wavelet-Homotopy Method. More orthogonal wavelet function will be applied in the framework of HAM. Unified strategy for nonlinear boundary value problem subjected to complex irregular boundary will be developed and attempted to generalize to the initial boundary value problems. Higher numerical accuracy connection coefficients database of compactly supported orthogonal wavelet will be established to overcome numerical sensitivity of iterating matrices. We will finally provide an effective tool to solve strongly nonlinear problems in mechanics science and ocean engineering.

非线性微分方程的高精度求解是力学与海洋工程问题定量分析中的一个关键瓶颈性难题。围绕其有效解决途径开展研究，具有重要的科学意义。小波同伦方法继承了同伦分析方法强大的求解强非线性问题能力和小波高精度逼近性质，成为求解非线性问题有效的重要数值方法。然而该方法目前只在矩形板大挠度弯曲和稳态方腔流动求解中得到应用，处在初步发展阶段，基函数局限于广义正交Coiflets小波，且小波和同伦方法结合局限于Galerkin方法，计算区域局限于如矩形板和方腔的正则方形区域。本项目拟将配点法和同伦分析方法相结合，发展高效的小波同伦配点方法，具有较大理论创新性，将更多正交小波函数族应用于同伦分析方法框架，研究满足复杂非规则边界的非线性边值问题统一求解方法，并尝试推广到非线性初边值问题，建立数值精度更高的紧支撑正交小波连接系数数据库，为求解力学科学和海洋工程中强非线性问题提供有效的方法。

非线性微分方程的高精度求解是力学与海洋工程问题定量分析中的一个关键瓶颈性难题，围绕其有效解决途径开展研究，具有重要的科学意义。 本项目针对广义正交 Coiflets 小波同伦方法进行拓展研究，运用牛顿莱布尼兹公式和广义小波伽辽金方法，将变系数强非线性偏微分控制方程等价变换并小波逼近，克服了变系数造成的复杂小波连接系数求解难题。 基于高阶泰勒级数展开，增加了边界插值矩阵的阶数，有效提高了广义正交 Coiflets 小波的逼近精度并建立了数值精度更高的紧支撑正交小波连接系数数据库。在水动力学领域研究了双扩散混合对流中二元纳米流体的多物理定常流场流动问题，给出了具有较大物理参数范围的复合场(流场、浓度场和温度场)小波级数解，研究了不同双扩散浮力系数、热纳米流体和热溶质Lewis数、热导系数等参数对复杂流动的影响。在板壳非线性力学领域研究了非线性弹性基支撑下任意轴向和横向约束的圆薄板与变厚度正交同(异)性矩形薄板、湿热与机械载荷联合作用下楔形板的大挠度几何非线性问题，给出了任意厚度分布下薄板变形挠度的小波级数解；在舰船结构领域提出了一种全新的等效正交板法，用于分析由双层面板与梁、腹板正交组合而成的双层底船体结构，采用等效变刚度法表征了夹层高度不均匀的弯曲特性，基于正交各向异性板大挠度理论对结构几何非线性弯曲进行了分析，与有限元结果对比验证了所得小波级数解具有较好的精度。本项目所发展的小波-同伦方法计算效率好且精度高，能有效处理强非线性，为力学科学和海洋工程中非线性问题提供了有效的求解方法。