Delve into the complexities of soil$-$structure interaction $($SSI$)$ and discuss how it influences the design and analysis of foundations in civil engineering. Consider the various factors that contribute to SSI, such as soil type, loading conditions, and structural characteristics. Explain the theoretical frameworks and computational models used to simulate SSI, highlighting the differences between traditional rigid foundation assumptions and more advanced flexible foundation models. Additionally, examine the role of SSI in dynamic analysis, particularly in the context of seismic loading, and discuss the methods used to account for soil damping and resonance effects. What challenges do engineers face when incorporating SSI into their designs, and how can these challenges be addressed through improved modeling techniques and field testing?