The paper presents a method for modeling arbitrary discontinuities in finite elements, including both function and derivative discontinuities. The approach uses a signed distance function to define discontinuity surfaces, allowing for efficient updating of these surfaces using level set methods. The method is based on a displacement Galerkin formulation and incorporates enrichment functions to handle discontinuities. The technique allows for the modeling of complex discontinuities such as cracks, joints, and shear bands without requiring remeshing. The method is applicable to a wide range of problems, including crack growth, journal bearings, non-bonded inclusions, and jointed rock masses. The approach is flexible, allowing for intersecting and branching discontinuities, and can be extended to vector functions and tangential discontinuities. The method is compared to global-local methods and is shown to be more efficient in terms of sparsity and computational cost. The paper also presents numerical examples demonstrating the effectiveness of the method in modeling discontinuities in various engineering applications. The method is particularly useful for problems involving moving discontinuities, as it avoids the need for remeshing and allows for the use of standard finite element meshes. The approach is validated through a series of numerical examples, showing its accuracy and robustness in modeling discontinuities in complex geometries.The paper presents a method for modeling arbitrary discontinuities in finite elements, including both function and derivative discontinuities. The approach uses a signed distance function to define discontinuity surfaces, allowing for efficient updating of these surfaces using level set methods. The method is based on a displacement Galerkin formulation and incorporates enrichment functions to handle discontinuities. The technique allows for the modeling of complex discontinuities such as cracks, joints, and shear bands without requiring remeshing. The method is applicable to a wide range of problems, including crack growth, journal bearings, non-bonded inclusions, and jointed rock masses. The approach is flexible, allowing for intersecting and branching discontinuities, and can be extended to vector functions and tangential discontinuities. The method is compared to global-local methods and is shown to be more efficient in terms of sparsity and computational cost. The paper also presents numerical examples demonstrating the effectiveness of the method in modeling discontinuities in various engineering applications. The method is particularly useful for problems involving moving discontinuities, as it avoids the need for remeshing and allows for the use of standard finite element meshes. The approach is validated through a series of numerical examples, showing its accuracy and robustness in modeling discontinuities in complex geometries.