An improved stress recovery technique for the unfitted finite element analysis of discontinuous gradient fields

Stress analysis is an all-pervasive practice in engineering design. With displacement-based finite element analysis, directly-calculated stress fields are obtained in a post-processing step by computing the gradient of the displacement field—therefore less accurate. In enriched finite element analysis (EFEA), which provides unprecedented versatility by decoupling the finite element mesh from material interfaces, cracks, and structural boundaries, stress recovery is further aggravated when such discontinuities get arbitrarily close to nodes of the mesh; the presence of small area integration elements often yields overestimated stresses, which could have a detrimental impact on nonlinear analyses (e.g., damage or plasticity) since stress concentrations are just a nonphysical numerical artifact. In this article, we propose a stress recovery procedure for enhancing the stress field in problems where the field gradient is discontinuous. The formulation is based on a stress improvement procedure (SIP) initially proposed for low-order standard finite elements. Although generally applicable to all EFEA, we investigate the technique with the Interface-enriched Generalized Finite Element Method and compare the procedure to other post-processing smoothing techniques. We demonstrate that SIP for EFEA provides an enhanced stress field that is more accurate than directly-calculated stresses—even when compared with standard FEM with fitted meshes.Structural Optimization and Mechanic

The Discontinuity-Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity-Enriched Finite Element Method (DE-FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction-free and cohesive crack examples. We show that DE-FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.Structural Optimization and MechanicsApplied Mechanic

Automated discrete element method calibration using genetic and optimization algorithms

This research aims at developing a universal methodology for automated calibration of microscopic properties of modelled granular materials. The proposed calibrator can be applied for different experimental set-ups. Two optimization approaches: (1) a genetic algorithm and (2) DIRECT optimization, are used to identify discrete element method input model parameters, e.g., coefficients of sliding and rolling friction. The algorithms are used to minimize the objective function characterized by the discrepancy between the experimental macroscopic properties and the associated numerical results. Two test cases highlight the robustness, stability, and reliability of the two algorithms used for automated discrete element method calibration with different set-ups.Transport Engineering and LogisticsStructural Optimization and Mechanic

Phononic crystals’ band gap manipulation via displacement modes

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.Computational Design and Mechanic

Multi-objective design of 3D phononic crystal waveguide by design space trimming

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.Computational Design and Mechanic

On tailoring fracture resistance of brittle structures: A level set interface-enriched topology optimization approach

We propose a fully immersed topology optimization procedure to design structures with tailored fracture resistance under linear elastic fracture mechanics assumptions for brittle materials. We use a level set function discretized by radial basis functions to represent the topology and the Interface-enriched Generalized Finite Element Method (IGFEM) to obtain an accurate structural response. The technique assumes that cracks can nucleate at right angles from the boundary, at the location of enriched nodes that are added to enhance the finite element approximation. Instead of performing multiple finite element analyses to evaluate the energy release rates (ERRs) of all potential cracks—a procedure that would be computationally intractable—we approximate them by means of topological derivatives after a single enriched finite element analysis of the uncracked domain. ERRs are then aggregated to construct the objective function, and the corresponding sensitivity formulation is derived analytically by means of an adjoint formulation. Several numerical examples demonstrate the technique's ability to tailor fracture resistance, including the well-known benchmark L-shaped bracket and a multiple-loading optimization problem for obtaining a structure with fracture resistance anisotropy.Structural Optimization and Mechanic

An engineering perspective to the virtual element method and its interplay with the standard finite element method

In this manuscript we thoroughly study the behavior of the virtual element method (VEM) in the context of two-dimensional linear elasticity problems for an engineering audience versed in standard FEM. Through detailed convergence studies we show the accuracy and the convergence rates recovered by VEM, and we compare them to those obtained by the h- and p-versions of the finite element method (FEM). We also demonstrate the mixability between FEM and VEM; in particular, applications of VEM for coupling non-conforming discretizations and for local refinement are presented, showing higher versatility compared to FEM. Computer implementation aspects in displacement-based finite element codes are thoroughly explained, remarking on the main differences with respect to standard FEM.Structural Optimization and Mechanic

Phononic crystals for suppressing crosstalk in ultrasonic flowmeters

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.Computational Design and Mechanic

On the stability and interpolating properties of the Hierarchical Interface-enriched Finite Element Method

The Hierarchical Interface-enriched Finite Element Method (HIFEM) is a technique for solving problems containing discontinuities in the field gradient using finite element meshes that do not conform (match) the domain morphology. The method is suitable for analyzing problems with complex geometries or when such geometry is not known a priori. Contrary to the eXtended/Generalized Finite Element Method (X/GFEM), enrichments are placed on nodes created along interfaces, and a recursive enrichment strategy is used to resolve multiple discontinuities crossing single elements. In this manuscript we rigorously study the approximating properties and stability of HIFEM. A study on the enrichments’ polynomial order shows that the formulation does not pass the patch test as long as enrichments do not replicate the approximating properties of partition of unity shape functions. Regarding stability, we show that condition numbers of system matrices grow at the same rate as those of standard FEM—and without requiring a preconditioner. This intrinsic stability is accomplished by means of a proper construction of enrichment functions that are properly scaled as interfaces approach mesh nodes. We further demonstrate that, even without scaling, using a simple preconditioner recovers stability. The method's stability is further demonstrated on the modeling of challenging thermal and mechanical problems with complex morphologies.Structural Optimization and Mechanic

Analytical characterization of the dynamic response of viscoelastic metamaterials

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.Computational Design and Mechanic