Metal additive manufacturing is a high-tech way to create complex parts in different industries. It involves adding new layers of material, like welding. The challenge is dealing with how solid, liquid, and heat interact, which is tricky for current computer methods. In our study, we suggest a method that combines these interactions in one approach. We use a technique called convected particle domain interpolation to track materials precisely. To make it efficient, we use a multi-grid technique. We test our method against existing data and find it's a powerful way to simulate the metal additive manufacturing process, helping us understand and improve it.