The implicit boundary finite-element method avoids the need for generating conforming mesh by using structured mesh for the analysis. A structured mesh (also referred to here as a grid) is a nonconforming mesh that is made up of regular shaped elements (rectangles for 2-D or cuboids for 3-D) and is easier to generate than traditional finite-element mesh. The geometry of the analysis domain is represented using equations that are independent of the grid. Essential boundary conditions are applied using solution structures that are constructed using approximate step functions of the boundary such that these boundary conditions are guaranteed to be enforced. A variety of interpolation functions and approximations such as B-splines can be used with this approach. Examples with known analytical solutions are used to validate the approach and to study convergence of elements using Lagrange interpolation and uniform B-spline approximation schemes.