Enriched finite element methods (e-FEMs) have become a popular choice for modeling problems containing material discontinuities (e.g., multi-phase materials and fracture). The main advantage as compared to the standard finite element method (FEM) remains the versatility in the choice of discretizations, since e-FEMs resolve discontinuities by completely decoupling them from the finite element mesh. However, modeling complex kinematics such as branching and merging of discrete cracks remains challenging. This article extends previous research on the Discontinuity-Enriched Finite Element Method (DE-FEM) for simulating quasi-static crack propagation in brittle materials. In DE-FEM enrichments are added to nodes created directly along discontinuities. Most notably, we demonstrate DE-FEM can resolve complex kinematics, namely the modeling of multiple cracks propagating and merging—and with a straightforward computer implementation. We validate the formulation with experimental results carried out on a compact tension specimen. Other numerical examples show the capability of DE-FEM in capturing crack paths similar to those observed in the literature.

Nonlinear dynamic simulations of mechanical resonators have been facilitated by the advent of computational techniques that generate nonlinear reduced order models (ROMs) using the finite element (FE) method. However, designing devices with specific nonlinear characteristics remains inefficient since it requires manual adjustment of the design parameters and can result in suboptimal designs. Here, we integrate an FE-based nonlinear ROM technique with a derivative-free optimization algorithm to enable the design of nonlinear mechanical resonators. The resulting methodology is used to optimize the support design of high-stress nanomechanical Si ₃N ₄ string resonators, in the presence of conflicting objectives such as simultaneous enhancement of Q-factor and nonlinear Duffing constant. To that end, we generate Pareto frontiers that highlight the trade-offs between optimization objectives and validate the results both numerically and experimentally. To further demonstrate the capability of multi-objective optimization for practical design challenges, we simultaneously optimize the design of nanoresonators for three key figure-of-merits in resonant sensing: power consumption, sensitivity and response time. The presented methodology can facilitate and accelerate designing (nano) mechanical resonators with optimized performance for a wide variety of applications. (Figure presented.)

We extend the Discontinuity-Enriched Finite Element Method (DE-FEM) to simulate intersecting discontinuities, such as those encountered in polycrystalline materials, multi-material wedge problems, and branched cracks. The proposed hierarchical enrichment functions capture weak and strong discontinuities at junctions within a single formulation. Several numerical applications to branched cracks and polycrystalline microstructures under both thermal and mechanical loads are presented to demonstrate the proposed method. Results indicate that DE-FEM can accurately capture complex discontinuous primal and gradient fields and attain convergence rates comparable to those of standard FEM using fitted meshes. The main advantages of DE-FEM equipped with the proposed junction enrichment functions lie in the method's ability to model intersecting discontinuities using meshes that are completely decoupled from them and its robustness in reproducing correct displacement and strain jumps across them, as demonstrated by a patch test. This work thus highlights the potential of DE-FEM for applications to problems characterized by the presence of multiple intersecting discontinuities, posing a valid alternative to traditional FEM and eXtended/Generalized Finite Element (X/GFEM) Methods.

The quality factor (Q factor) of nanomechanical resonators is influenced by geometry and stress, a phenomenon called dissipation dilution. Studies have explored maximizing this effect, leading to softly-clamped resonator designs. This paper proposes a topology optimization methodology to design two-dimensional nanomechanical resonators with high Q factors by maximizing dissipation dilution. A formulation based on the ratio of geometrically nonlinear to linear modal stiffnesses of a prestressed finite element model is used, with its corresponding adjoint sensitivity analysis formulation. Systematic design in square domains yields geometries with comparable Q factors to literature. We analyze the trade-offs between resonance frequency and quality factor, and how these are reflected in the geometry of resonators. We further apply the methodology to optimize a resonator on a full hexagonal domain. By using the entire mesh—i.e., without assuming any symmetries—we find that the optimizer converges to a two-axis symmetric design comprised of four tethers.

Clamp-on ultrasonic flowmeters suffer from crosstalk—i.e., measurement errors due to the interference of signals generated in solid regions and solid–fluid interfaces with the required signal from the fluid. Although several approaches have been proposed to alleviate crosstalk, they only work in specific ranges of flow rates and pipe diameters, and some also introduce additional issues. We propose a novel clamp-on system design where the transmitting and receiving wedges are embedded with directional noise filtering mechanisms based on phononic crystals (PnCs) possessing directional band gaps (DBGs). PnCs are artificial materials consisting of periodic structures arrayed in a matrix medium exhibiting band gaps – i.e., frequency ranges where waves are attenuated – due to Bragg scattering. DBGs enable PnCs to propagate waves in specific directions while suppressing them in other directions. By guiding the input signal through the transmitting wedge to the wall, we minimize the generation of noise signals due to secondary reflections within the wedge. Similarly, by using the directionality of the DBG PnC in the receiver, we limit the effects of noise signals (that arrive in different directions) in the receiver. We numerically verify the DBG PnC embedded wedges’ performance by comparing wave propagation aspects of the PnC embedded clamp-on system with a standard clamp-on device. To that end, we develop accurate wave propagation models based on the Discontinuous Galerkin finite element method. By incorporating DBG PnCs into the wedges, we obtain about 20 dB increase in the signal-to-noise ratio compared to the clamp-on system with standard wedges.

Continuous fiber fused filament fabrication (CF4) is a layer-by-layer additive manufacturing technique that deposits continuous fiber fused filaments (CFFFs) with a significant in-plane variation of the fiber trajectory, thereby offering great flexibility in fabricating variable-stiffness composite laminates (VSCLs). We introduce a topology optimization method for the design of additively manufactured VSCLs made of overlapping, fiber-reinforced bars. The proposed method is based on geometry projection (GP) techniques, whereby the bars are represented by high-level geometric primitives. As in other GP techniques, this high-level parameterization is mapped onto a fixed structured finite element mesh for conducting analysis, as in density-based topology optimization techniques. However, unlike previous GP techniques that have demonstrated their applicability in designing structures as assemblies of individual fiber-reinforced components, this work focuses on the design of composite structures that adhere to CF4 manufacturing processes. Therefore, we first formulate a material interpolation scheme that better captures the stiffness at the composite's joints obtained from bar overlaps as a stack. Second, the proposed material interpolation employs composite laminate theory to capture the in-plane and out-of-plane behavior of the structure. Third, to produce designs that conform to the CF4 process, we also proposed a novel length constraint formulation in the form of penalization on the projection scheme, which ensures a minimum length for all the bars. This minimum length limit does not require adding a constraint to the optimization problem. The efficacy and efficiency of the proposed method are demonstrated by a series of compliance minimization problems with in-plane and/or out-of-plane loading. The methodology is also applied to the design of a displacement inverter compliant mechanism.

This paper addresses the topology optimization of thermocouples for cooling applications, considering stress constraints to enhance reliability under service loads. We provide a first approach to derive sensitivities using SIMP (solid isotropic material with penalization) for thermo-electro-mechanical systems with temperature-dependent material properties. The proposed formulation decouples the thermoelectrical system from the mechanical degrees of freedom reducing computational memory usage from a fully coupled approach. The study focuses on the formulation of thermocouples for cooling applications using the Peltier effect, which considers electrical power limits, electrical working points, and material stress thresholds. Furthermore, while the thermoelectrical problem does not show the need for filtering techniques, including the mechanical degrees of freedom, we show that we recover undesirable porous optimized designs. We provide 2D thermocouple example optimizations with geometries and boundary conditions based on a practical case for the implementation of thermoelectric coolers in the Minimum Ionizing Particle Timing Detector (MTD) at CERN. The optimizations are performed with increased complexity, including the unfiltered thermoelectrical and thermo-electro-mechanical problems and a Helmholtz-filtered examples. The optimizations are compared with constant and nonlinear material properties with temperature and with respect to the consideration of air-conductance losses within the devices. Although more efficient topologies can be achieved without the need for volume constraints, we include an example with a constraint of 60% volume to understand its effect on the design and provide a methodology to reduce semiconductor-associated costs at lower efficiency costs. Finally, we explore the same formulation in 3D. The results provide guidelines for manufacturing compliant thermocouples, increasing their reliability without decreasing efficiency.

The computational analysis of nanophotonic devices is usually carried out via the standard finite element method (FEM). However, FEM requires meshes that are fitted to the devices’ boundaries, so making changes to the geometry (and thus the mesh) results in an inefficient process at best. Such an approach is therefore at odds when conducting design, which requires the analysis of multiple device geometries until reaching a satisfactory solution. Computational design tools such as topology optimization are often used, but the use of density-based representations of geometry inevitably leads to other issues—e.g., pixelized fuzzy boundaries with “gray material” (that does not correspond to dielectric nor vacuum) have an adverse effect on the devices’ interaction with electromagnetic waves. In this paper we propose an interface-enriched generalized finite element method (IGFEM) for the analysis of two-dimensional electromagnetic scattering and eigenvalue problems. IGFEM enables the use of finite element meshes that are completely decoupled from the problem's geometry. The analysis procedure is further coupled to a level set description of topology, resulting in a versatile enriched approach to topology optimization; this level set-based interface-enriched topology optimization procedure is devoid of the issues mentioned above regarding density-based methods, and yields crisp “black-and-white” designs that are devoid of jagged fuzzy edges. We first demonstrate that the analysis procedure achieves the same convergence rate as that of standard FEM using geometry-fitted meshes. We then compare the convergence properties of IGFEM with Nitsche's method on a problem containing an embedded straight interface. Finally, we conduct topology optimization for designing both a 2-D metalens and a 2-D reflector, maximizing their ability to focus light onto a target point.

Although strain engineering and soft-clamping techniques for attaining high Q-factors in nanoresonators have received much attention, their impact on nonlinear dynamics is not fully understood. In this study, we show that nonlinearity of high-Q Si₃N₄ nanomechanical string resonators can be substantially tuned by support design. Through careful engineering of support geometries, we control both stress and mechanical nonlinearities, effectively tuning nonlinear stiffness of two orders of magnitude. Our approach also allows control over the sign of the Duffing constant resulting in nonlinear softening of the mechanical mode that conventionally exhibits hardening behavior. We elucidate the influence of support design on the magnitude and trend of the nonlinearity using both analytical and finite element-based reduced-order models that validate our experimental findings. Our work provides evidence of the role of soft-clamping on the nonlinear dynamic response of nanoresonators, offering an alternative pathway for nullifying or enhancing nonlinearity in a reproducible and passive manner.

Current design methods for flexure (or compliant) mechanisms regard stress as a secondary, limiting factor. This is remarkable because stress is also known as a useful design parameter. In this paper we propose the Stress And Geometry (STAGE) method, to design the geometry of a flexure mechanism together with a desired stress field. From this design, the stress-free to-be-fabricated geometry is computed using the inverse finite element method. To demonstrate the potential of the method, the geometry of the well-known crossed-flexure pivot is taken as example. We first show how this mechanism can be redesigned for the same functional geometry with various internal stresses. This results for a specific choice of stress field in a design of a crossed-flexure pivot with 23% lower peak stresses during motion as compared to the known designs, for a ±45° rotation. We then present a second example, of a Folded Leaf Spring (FLS). With a parameter sweep the optimal stress field is calculated, showing a peak stress reduction of 28% during motion. This result was validated with an experiment, showing a normalized mean absolute error of 5.5% between experiment and theory. With a second experiment it was verified that the functional geometry of the FLS with internal stresses was equal to the one without internal stresses, with geometric deviations smaller than half the thickness of the flexures.

Inorganic scintillators often use exotic, expensive materials to increase their light yield. Although material chemistry is a valid way to increase the light collection, these methods are expensive and limited to the material properties. As such, alternative methods such as the use of specific reflective coatings and crystal optical shapes are critical for the scintillator crystal design procedure. In this paper, we explore the modeling of a scintillator and silicon-photomultiplier (SiPM) assembly detector using GEANT4. GEANT4, an open-source software for particle–matter interaction based on ray-tracing, allows the modeling of a scintillator-based detector while offering methods to simplify and study the computational requirements for a precise calculation of the light collection. These studies incorporate two different geometries compatible with the barrel timing layer (BTL) particle detector that is being built for the compact muon solenoid (CME) experiment at CERN. Furthermore, the geometry of our model is parameterized using splines for smoother results and meshed using GMSH to perform genetic numerical optimization of the crystal shape through genetic algorithms, in particular non-dominated sorting genetic algorithm II (NGSAII). Using NSGA-II, we provide a series of optimized scintillator geometries and study the trade-offs of multiple possible objective functions including the light output, light collection, light collection per energy deposited, and track path length. The converged Pareto results according to the hypervolume indicator are compared to the original simplified design, and a recommendation towards the use of the light collection per energy deposition and track path length is given based on the results. The results provide increases in this objective of up to 18% for a constant volume for a geometry compatible with the current design of the BTL detector.

Underwater noise resulting from the monopile driving process can cause severe damage to marine wildlife, such as hearing injury, behavioral disturbance, or even death. Although current noise-attenuation techniques used in this process have shown a significant noise reduction at high frequency ranges, mitigating low-frequency noise is still extremely challenging. To address the problem, here we propose an elastic metamaterial-based structure composed of single-phase resonant structures. The proposed structure, which we call a meta-interface, is introduced between the monopile and the hammer and is used to remove energy from the input signal associated with high noise levels. To that end, we first identify the frequency ranges associated with high sound pressure levels, which were shown to be related to the monopile's eigenmodes. Then we design the meta-interface's periodic unit cells so that the elastic/acoustic waves at identified frequency ranges are attenuated. A meta-interface is then realized by replicating the unit cell along the monopile wall (matching the thickness) to form a ring-shaped layer, and then by stacking up these concentric layers. A frequency analysis of the pile driving system with the meta-interface shows that the new noise levels attain a significant attenuation in frequency ranges lower than 1000Hz. This demonstrates a novel solution for the low-frequency underwater noise issue during the hammering of offshore monopiles.

Fundamentals of Enriched Finite Element Methods provides an overview of the different enriched finite element methods, detailed instruction on their use, and their real-world applications, recommending in what situations they are best implemented. It starts with a concise background on the theory required to understand the underlying principles behind the methods before outlining detailed instruction on implementation of the techniques in standard displacement-based finite element codes. The strengths and weaknesses of each are discussed, as are computer implementation details, including a standalone generalized finite element package, written in Python. The applications of the methods to a range of scenarios, including multiphase, fracture, multiscale, and immersed boundary (fictitious domain) problems are covered, and readers can find ready-to-use code, simulation videos, and other useful resources on the companion website of the book.

Ultrasonic flowmeters face unique challenges since, in addition to withstanding high fluid pressures, they have to avoid crosstalk, which is the interaction of the signals traveling through the fluid and the solid pipe. To avoid the crosstalk, which leads to poor accuracy or complete loss of the required signal, we develop a mounting mechanism based on phononic crystals (PnCs), which are artificial periodic materials possessing band gaps (BGs) due to Bragg scattering. These PnC structures should also possess high mechanical strength to sustain the fluid pressure. Designing PnCs for such applications is challenging as the BG width and the resistance to mechanical loading are conflicting objectives. To circumvent this, we propose a step-by-step design procedure to optimize both mechanical strength and wave attenuation performance of a single-phase 3D PnC waveguide using parametric sweeping and sensitivity analysis. We use finite element analysis (FEA) to characterize the behavior of the periodic unit cell and the waveguide. Since accurate dynamic FEA at high frequencies is computationally demanding, we develop surrogate models at different levels of the design process. We also consider additive manufacturing aspects in the design procedure, which we validate by 3D-printing the final design and measuring the parameters via computer tomography.

Heat pumping through thermoelectric devices has many advantages over traditional cooling. However, their current efficiency is a limiting factor in their implementation. In this paper, we approach the non-convex topology optimization of thermoelectrical elements for cooling applications through the method of moving asymptotes (MMA) to improve their cooling capabilities per watt usage. The optimization problem is defined for a given power budget, aiming for the minimum temperature with a known heat pumping need. The introduction of power as a constraint justifies the introduction of the voltage gradient across the thermocouple as a design variable to maintain the thermoelectrical device in its optimum power-to-heat extraction ratio. To better understand the convergence of this non-convex problem, we present a two-variable analytical thermoelectric optimization model. This example provides information on how to select the penalty parameters used to scale the three material coefficients involved in the problem to obtain lower objective values and better convergence using MMA. The analytical model shows the non-convexity of the problem and provides the recommendation to use penalization coefficients of the form p_k=p_σ>p_α=1 for the thermal conductivity, electrical conductivity, and Seebeck coefficients. We tested these penalization coefficients through optimizations of a model based on the 1MC10-031 commercial thermoelectric-cooler (TEC) using the finite element method (FEM). These penalization coefficients provided local minima without the need for volume constraints. With this procedure, we found designs that provided temperatures close to 10 degrees lower using 60% less semiconductor material volume compared to the initial design.

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.

The band-gap frequencies of elastic metamaterials are ideally determined by a metamaterial architecture; yet, in practical situations, are often dependent on the material damping in their constituent(s). The analysis of viscoelastic metamaterials requires however substantial computational resources and, except for oversimplified cases, is solely done numerically. Here, we propose an analytical procedure based on the spectral element method (SEM) to analyze bulk metamaterials with viscoelastic damping as continuous systems. Due to intrinsic limitations of the SEM to deal with complex geometries, we develop a procedure to build an approximate model based on SEM frame elements. The viscoelastic behavior is included by means of complex viscoelasticity moduli expressed by the generalized Maxwell mechanical model. We validate this approach by analyzing metamaterial plates and verify the findings experimentally. We demonstrate that our SEM-based analytical model can accurately capture wave transmission around the first band-gap frequencies. Therefore, our extension of the SEM approach to analyze three-dimensional meta-structures is promising to characterize wave propagation in realistic viscoelastic structures (with any type of linear viscoelastic behavior) in an accurate and computationally efficient way.

Phononic crystal band gaps (BGs), which are realized by Bragg scattering, have a central frequency and width related to the unit cell's size and the impedance mismatch between material phases. BG tuning has generally been performed by either trial and error or by computational tools such as topology optimization. In either case, understanding how to systematically change the design for a particular band structure is missing. This paper addresses this by closely studying the displacement modes within the wavebands that are responsible for the BG. We look at the variation in different displacement modes due to the changes in the geometry and correlate these changes to their corresponding band structures. We then use this insight to design the unit cell for a particular application, for instance, for generating partial BGs.

In recent years, the Q-factor of Si 3 N 4 nanomechanical resonators has significantly been increased by soft-clamping techniques using large and complex support structures. To date, however, obtaining similar performance with smaller supports has remained a challenge. Here, we make use of torsion beam supports to tune the Q-factor of Si 3 N 4 string resonators. By design optimization of the supports, we obtain a 50% Q-factor enhancement compared to the standard clamped-clamped string resonators. By performing experimental and numerical studies, we show that further improvement of the Q-factor is limited by a trade-off between maximizing stress and minimizing torsional support stiffness. Thus, our study also provides insight into dissipation limits of high-stress string resonators and outlines how advanced designs can be realized for reaching ultimate f 0 × Q product while maintaining a small footprint.

Ultrasonic flowmeters that use transit-time ultrasonic transducers face measurement errors due to 'crosstalk,' whereby the working signal travels through the pipe wall and couplings, interfering with the signal from the fluid. Although various procedures have been proposed to solve the issue of crosstalk, they're limited to low-frequency ranges, or they are not effective in high-pressure environments. We propose a mounting mechanism based on a single-phase 3-D phononic crystal (PnC) waveguide that can mitigate crosstalk at high frequencies (megahertz range) and thus improve the flowmeters' measurement accuracy. PnCs are artificial materials consisting of periodically arranged scatterers thereby showing bandgaps (BGs) - ranges of frequencies where elastic/acoustic waves are attenuated - due to Bragg scattering. We design PnC wave filters by engineering the BG frequency range to the working signal of the ultrasonic flowmeter. We then fabricate the waveguide using additive manufacturing and connect it between the transducer and the pipe wall. Transient ultrasonic experiments show that transducers with PnC mountings attain a 40 dB crosstalk reduction in comparison with a standard transducer mounting configuration.