We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we compute localized bounded states at the lower end of the spectrum. We present several deep network architectures that predict the localization of bounded states from a sample of a potential. For tackling higher dimensional problems, we consider a class of physics-informed deep dense networks. In particular, we focus on the interpretability of the proposed approaches. Deep network is used as a general reduced order model that describes the nonlinear connection between the potential and the ground state. The performance of the surrogate reduced model is controlled by an error estimator and the model is updated if necessary. Finally, we present a host of experiments to measure the accuracy and performance of the proposed algorithm.

This paper presents an algorithm for the fully automatic mesh generation for the finite element analysis of ships and offshore structures. The quality requirements on the mesh generator are imposed by the acceptance criteria of the classification societies as well as the need to avoid shear locking when using low degree shell elements. The meshing algorithm will be generating quadrilateral dominated meshes (consisting of quads and triangles) and the mesh quality requirements mandate that quadrilaterals with internal angles close to 90° are to be preferred. The geometry is described by a dictionary containing points, rods, surfaces, and openings. The first part of the proposed method consists of an algorithm to automatically clean the geometry. The corrected geometry is then meshed by the frontal Delaunay mesh generator as implemented in the gmsh package. We present a heuristic method to precondition the cross field of the fronatal quadrilateral mesher. In addition, the influence of the order in which the plates are meshed will be explored as a preconditioning step.

We present an algorithm for the fully automatic generation of a class-compliant mesh for ship structural analysis. Our algorithm is implemented as an end-to-end solution. It starts from a description of a geometry and produces a class conforming surface mesh as a result. The algorithm consists of two parts, the automatic geometry refinement and the preconditioned Delaunay frontal quad dominant mesh generator. A geometry is described by a dictionary of elements and it contains points, rods, plates, and openings. A dictionary can contain modeling errors such as unintended overlaps or an unintended loss of connectivity between elements. The main contribution of the paper is the automatic geometry refinement algorithm and the virtual stiffener procedure designed to control the local mesh orientation of a marching front meshing algorithm. The geometry refinement algorithm guarantees that the output dictionary will be such that intersections of the boundary edges of plates are guaranteed to be nodes of any mesh generated by tessellating such geometry. The algorithm is implemented in Python, using the open-source Gmsh system together with the Open CASCADE kernel which is used to implement the automatic geometry refinement. We present several benchmark models from an engineering practice to illustrate our claims as well as to benchmark the efficiency of the various stages of the processing pipeline.