Phononic crystals can be designed to show band gaps—ranges of frequencies whose propagation is strongly attenuated in the material. In essence, their working principle is based on destructive interference of waves reflecting from the periodic arrangement of material interfaces (i.e., Bragg scattering). Consequently, capturing accurately the behavior at material interfaces requires appropriate numerical modeling and computational design techniques. However, the commonly used density-based representation in popular topology optimization methods results in a diffuse staircased boundary. The heavily refined finite element meshes required to compensate for this boundary description results in exceedingly large and expensive optimization problems. In this paper, we demonstrate the adverse effect of the density-based boundary description. Furthermore, we propose a level set-based topology optimization procedure with an enriched finite element method that shows improved performance when compared to the density-based approach.

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The published article contains a mistake in the notation of the strain–displacement matrix (Formula presented.) in Voigt notation concerning the isoparametric mapping and in the notation of the corresponding derivatives. As the mistakes relate only to the notation, and the implementation in the code was done correctly, they bear no consequences for the remainder of the manuscript and the results presented therein. The (Formula presented.) matrix defined inline after Eq. (11) on page 5 (repeated on pages 8 and 17) should be written as (Formula presented.) for an enriched element with n original shape functions and m enrichment functions. The differential operator (Formula presented.) in Eq. 12 should then have been defined in Voigt notation and in global coordinates as (Formula presented.) for elastostatics in 2-D and 3-D, respectively, and (Formula presented.) for heat conductivity in 2-D and 3-D, respectively. The derivatives in global coordinates are computed from the derivatives in local coordinates as (Formula presented.) for standard and enriched shape functions, respectively, where (Formula presented.) is the Jacobian of the intersected original element and (Formula presented.) is the Jacobian of the integration element. Furthermore, the derivative of (Formula presented.) with respect to the enriched node location (Formula presented.) in Equation 28 should be written as (Formula presented.) Note that the terms in (Formula presented.) corresponding to the standard shape functions are zero (this equation replaces Eq. 29 of the manuscript): (Formula presented.) as the Jacobian of the parent element is not influenced by the enriched node location, and the derivatives of the standard shape functions with respect to local coordinates of the parent element are constant. The components in (Formula presented.) corresponding to the enrichment functions are computed using (this equation replaces Eq. 30 in the manuscript) (Formula presented.) where the second term is zero because the derivatives of the enrichment function with respect to local coordinates are constant and independent of the enriched node location in global coordinates.

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Real-world structural optimisation problems involve multiple loading conditions and design constraints, with responses typically depending on states of discretised governing equations. Generally, one uses gradient-based nested analysis and design approaches to solve these problems. Herein, solving both physical and adjoint problems dominates the overall computational effort. Although not commonly detected, real-world problems can contain linear dependencies between encountered physical and adjoint loads. Manually keeping track of such dependencies becomes tedious as design problems become increasingly involved. This work proposes using a Linear Dependency Aware Solver (LDAS) to detect and exploit such dependencies. The proposed algorithm can efficiently detect linear dependencies between all loads and obtain the exact solution while avoiding unnecessary solves entirely and automatically. Illustrative examples demonstrate the need and benefits of using an LDAS, including a run-time experiment.

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We propose an enriched finite element formulation to address the computational modeling of contact problems and the coupling of non-conforming discretizations in the small deformation setting. The displacement field is augmented by enriched terms that are associated with generalized degrees of freedom collocated along non-conforming interfaces or contact surfaces. The enrichment strategy effectively produces an enriched node-to-node discretization that can be used with any constraint enforcement criterion; this is demonstrated with both multi-point constraints and Lagrange multipliers, the latter in a generalized Newton implementation where both primal and Lagrange multiplier fields are updated simultaneously. We show that the node-to-node enrichment ensures continuity of the displacement field—without locking—in mesh coupling problems, and that tractions are transferred accurately at contact interfaces without the need for stabilization. We also show the formulation is stable with respect to the condition number of the stiffness matrix by using a simple Jacobi-like diagonal preconditioner.

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In this work, an object-oriented geometric engine is proposed to solve problems with discontinuities, for instance, material interfaces and cracks, by means of unfitted, immersed, or enriched finite element methods (FEMs). Both explicit and implicit representations, such as geometric entities and level sets, are introduced to describe configurations of discontinuities. The geometric engine is designed in an object-oriented way and consists of several modules. For efficiency, a (Formula presented.) -d tree data structure that partitions the background mesh is constructed for detecting cut elements whose neighbors are found by means of a dual graph structure. Moreover, the implementation for creating enriched nodes, integration elements, and physical groups is described in detail, and the corresponding pseudo-code is also provided. The complexity and efficiency of the geometric engine are investigated by solving 2-D and 3-D discontinuous models. The capability of the geometric engine is demonstrated on several numerical examples. Topology optimization and problems with intersecting discontinuities are handled with enriched FEMs, where enriched discretizations obtained from the geometric engine are used for the analysis. Furthermore, polycrystalline structures that overlap with an unfitted mesh are considered, where integration elements are created so they align with grain boundaries. Another example shows that the Stanford bunny, which is discretized by a surface mesh with triangular elements, can be fully immersed into a 3-D background mesh. Finally, we share a list of main findings and conclude that the proposed geometric engine is general, robust, and efficient.

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Phononic crystals can be designed to have bandgaps---ranges of frequencies whose propagation through the material is prevented. They are therefore attractive for vibration isolation applications in different industries, where unwanted vibrations reduce performance. Yet, important steps are still to be made for the integration of phononic crystals into engineering practice. For instance, methods for large scale production are still in development. Furthermore, it is essential that design methods are established to enable the design of phononic crystals that meet all of the, often conflicting, requirements for practical applications. This thesis focuses on the latter challenge by proposing a computational design method for phononic crystals based on the combination of an advanced finite element method and level set-based topology optimization.@en

An immersed enriched finite element method is proposed for the analysis of phononic crystals (PnCs) with finite element (FE) meshes that are completely decoupled from geometry. Particularly, a technique is proposed to prescribe Bloch–Floquet periodic boundary conditions strongly on non-matching edges of the periodic unit cell (PUC). The enriched finite element formulation effectively transforms a periodic non-confirming discretization into an enriched node-to-node periodic discretizations where periodicity is enforced by any standard procedure. The enriched formulation is also used to describe the interior material interface. This completely eliminates the tedious process of generating matching or fitted meshes during the design process, as it allows changing the inclusion's geometry as well as the PnC's lattice type without changing the FE mesh. The proposed approach, which is used to analyze phononic crystals in 1-D, 2-D, and 3-D using structured meshes, exhibits the same performance as the standard finite element analysis on fitted meshes.

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Enriched finite element methods have gained traction in recent years for modeling problems with material interfaces and cracks. By means of enrichment functions that incorporate a priori behavior about the solution, these methods decouple the finite element (FE) discretization from the geometric configuration of such discontinuities. Taking advantage of this greater flexibility, recent studies have proposed the adoption of Non-Uniform Rational B-Splines (NURBS) to preserve the interfaces' exact geometries throughout the analysis. In this article, we investigate NURBS-based geometries in the context of the Discontinuity-Enriched Finite Element Method (DE-FEM) based on linear field approximations. While optimal convergence is retained for problems with weak discontinuities without singularities, representing exact geometry via NURBS does not yield noticeable improvements when extracting stress intensity factors of cracked specimens. For low-order elements, we conclude that the benefits of exact geometry representation do not outweigh the increased complexity in formulation and implementation. The choice of linear FEs hinders the accuracy of the proposed formulation, suggesting that its full potential may only be unleashed by increasing the field representation order.

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During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

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Generating matching meshes for finite element analysis is not always a convenient choice, for instance, in cases where the location of the boundary is not known a priori or when the boundary has a complex shape. In such cases, enriched finite element methods can be used to describe the geometric features independently from the mesh. The Discontinuity‐Enriched Finite Element Method (DE‐FEM) was recently proposed for solving problems with both weak and strong discontinuities within the computational domain. In this paper, we extend DE‐FEM to treat fictitious domain problems, where the mesh‐independent boundaries might either describe a discontinuity within the object, or the boundary of the object itself. These boundaries might be given by an explicit expression or an implicit level set. We demonstrate the main assets of DE‐FEM as an immersed method by means of a number of numerical examples; we show that the method is not only stable and optimally convergent but, most importantly, that essential boundary conditions can be prescribed strongly.@en

A new enriched finite element technique, named the Discontinuity-Enriched Finite Element Method (DE-FEM), was introduced recently for solving problems with both weak and strong discontinuities in 2-D. In this mesh-independent procedure, enriched degrees of freedom are added to new nodes collocated at the intersections between discontinuities and the sides of finite elements of the background mesh. In this work we extend DE-FEM to 3-D and describe in detail the implementation of a geometric engine capable of handling interactions between discontinuities and the background mesh. Several numerical examples in linear elastic fracture mechanics demonstrate the capability and performance of DE-FEM in handling discontinuities in a fully mesh-independent manner. We compare convergence properties and the ability to extract stress intensity factors with standard FEM. Most importantly, we show DE-FEM provides a stable formulation with regard to the condition number of the resulting system stiffness matrix.

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he numerical modeling of phononic crystals using the finite element method requires a mesh that accurately describes the geometric features. In an optimization setting, involving shape and/or topological changes, this implies that a new matching mesh needs to be generated in every design iteration. In this paper a mesh-independent description for both the interior and exterior boundaries of the periodic unit cell is proposed. A method is developed to apply Bloch-Floquet periodic boundary conditions to edges that are non-matching to the mesh. The proposed method is applied to a one-dimensional phononic crystal and is demonstrated to exhibit improved performance over the commonly used interface material averaging. We show that this method provides an accurate mesh-independent model.@en