Recent trends in computer science have emphasized the use of numerical optimization methodologies in physically-based simulations. By treating time-stepping problems as optimization targets and employing optimization techniques, both the accuracy and efficiency of solvers have been significantly improved. This thesis begins with an exploration of nonlinear optimization theory, then proceeds to present novel algorithms designed to simulate non-equilibrated viscoelastic and elastoplastic solids implicitly, even under large time steps. Collaborative efforts within the computer graphics community have enabled the realistic digital reproduction of various daily life materials, such as snow, sand, water, and jelly. These research breakthroughs have had profound implications across diverse fields, such as gaming, film production, fashion design, robotics, and mechanical engineering analysis. Among the myriad of applications, the concept of contact remains a vital factor influencing simulation results, which motivated us to refocus our research on this area. Free-slip contact, frictional contact, and adhesive contact each exhibit their unique characteristics when integrated with different numerical methods. For meshless methods, like the Material Point Method (MPM), the presence of a background grid confers advantages in automatic collision detection, although the contacting surface is only implicitly defined. Conversely, mesh-based methods, such as the Finite Element Method (FEM), require considerable effort to query exact primitive contacting pairs. Due to the strict definition of the mesh, intersections must be entirely eliminated to avoid introducing significant artifacts in later stages of the process. Notably, in addition to providing visual enjoyment with complex contact scenarios, our framework boasts many other functionalities, including the ability to create new shapes that mimic human craftsmanship and the capacity to align closely with real-world experimental results.