Physics-Informed Neural Networks (PINNs) and the Deep Energy Method (DEM) are two novel deep neural network-based methods for solving PDEs, where DEM works by lowering total potential energy, whereas PINNs minimise strong-form residuals [1]. Despite their promise, both approaches struggle to accurately represent the steep phase-field profiles in fracture and the sharp gradients and localised features in displacement and stress fields in stress-concentration problems[2, 3]. We propose I-EMM, a discrete energy-minimization framework that combines the foundation of DEM with isogeometric analysis (IGA). The method uses NURBS/B-spline with B´ezier extraction to discretize the displacement, phase field parameter and treats the resulting degrees of freedom as optimization variables. In this method, instead of assembling and solving the stiffness equation (Ku = f ), we treat the nodal values as optimization variables and minimize the total potential energy. We demonstrate the approach on standard 2D linear-elasticity benchmarks, including stress-concentration geometries, and assess accuracy against analytical solutions when available and IGA reference solutions otherwise. We further apply this framework to phase-field fracture problem under incremental displacement loading and validate both displacement and phase field predictions against IGA reference solutions. A parametric study is conducted to evaluate the sensitivity of the method to key modeling and numerical parameters such as refinement, quadrature, and length scale parameter settings. The results show that I-EMM provides accurate and robust solutions for linear elasticity and phase field fracture problems, with strong performance in resolving localised features, highlighting its potential as a practical energy-based alternative for challenging mechanics applications.