This study proposes a generalized framework to establish and maintain a comprehensive continuity and provide computational efficiency for solving topology optimization (TO) problems in structural mechanics by utilizing the technique of multi-patch isogeometric analysis (IGA). The continuity of the solution inside a patch is realized by taking the benefit of IGA to easily discretize the \(C^0\) and \(C^1\) continuous weak form of second and fourth-order differential equations respectively. The continuity of the solution between patches for fourth-order systems is established through a strong \(C^1\) coupling at patch boundaries, where new basis functions are constructed as a linear combination of existing \(C^0\) bases at the patch interfaces. A continuous density function (CDF) is then established by linearly combining the design vector and the isogeometric basis functions, providing a continuous smooth distribution of material for the TO in the design domain. Furthermore, an adaptive mesh refinement (AMR) approach is incorporated into the methodology to enhance computational efficiency by significantly reducing the degree’s-of-freedom and CPU time required to achieve the converged solution. Collectively, these integrated methodologies establish a robust and versatile framework capable of addressing a wide spectrum of structural mechanics problems, as demonstrated through numerical examples, with a particular emphasis on the challenges posed by fourth-order systems. The results highlight the continuity-preserving nature of the approach and achieve up to \(90\, \%\) reduction in degrees of freedom and elements, underscoring its computational efficiency and accuracy.
