<p>Topology optimization has been widely used in various engineering disciplines. Some topology optimization methods use partial elements to generate high-resolution structures while maintaining design freedom. Partial elements are elements with both solid and void phases that appear around the boundaries of designs. Due to their mixed nature, accurately estimating responses in partial elements is challenging. Existing methods to calculate responses and sensitivities in partial elements are either simple but inaccurate or prohibitively complicated. This paper uses a novel finite-element formulation for estimating properties of arbitrarily shaped partial elements, maintaining a balance between simplicity and accuracy. In this approach, partial elements are modeled as macro-elements containing micro-elements. This enables efficient calculation of stiffness matrices and sensitivity analysis of partial elements. This method can be readily integrated into existing topology optimization processes, allowing up-sampling of results obtained based on coarse mesh to finer mesh solutions with enhanced boundary resolutions. The effectiveness of this proposed approach has been validated through example problems, including compliance minimization, compliant mechanism design, and natural frequency maximization.</p>