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The practice of verification of shell finite element formulations is discussed from a variety of viewpoints: The provenance of benchmark target values, the suitability of benchmark problems for different classes of shell models, the importance of reporting limit values, and the completeness of the benchmark sets are critically examined. A set of benchmark problems that are most suitable across common shell finite element models is identified.

A simple procedure for estimating the uncertainty of estimates of true solutions to problems of deflection, stress concentrations, and force resultants in solid and structural mechanics is introduced. Richardson extrapolation is carried out on a dataset of samples from a sequence of four grids. Simple median-based statistical analysis is used to establish 95% confidence intervals. The procedure leads to simple calculations that deliver reasonably tight estimates of the true solution and confidence intervals.

A recently introduced representation by a set of Wang tiles—a generalization of the traditional Periodic Unit Cell-based approach—serves as a reduced geometrical model for materials with stochastic heterogeneous microstructure, enabling an efficient synthesis of microstructural realizations. To facilitate macroscopic analyses with a fully resolved microstructure generated with Wang tiles, we develop a reduced order modelling scheme utilizing pre-computed characteristic features of the tiles. In the offline phase, inspired by computational homogenization, we extract continuous fluctuation fields from the compressed microstructural representation as responses to generalized loading represented by the first- and second-order macroscopic gradients. In the online phase, using the ansatz of the generalized finite element method, we combine these fields with a coarse finite element discretization to create microstructure-informed reduced modes specific for a given macroscopic problem. Considering a two-dimensional scalar elliptic problem, we demonstrate that our scheme delivers less than 3% error in both the relative $$L_2$$ and energy norms with only 0.01% of the unknowns when compared to the fully resolved problem. Accuracy can be further improved by locally refining the macroscopic discretization and/or employing more pre-computed fluctuation fields. Finally, unlike standard snapshot-based reduced-order approaches, our scheme handles significant changes in the macroscopic geometry or loading without the need for recalculating the offline phase, because the fluctuation fields are extracted without any prior knowledge of the macroscopic problem.

Biological materials tested in compression, tension, and impact inspire designs for strong and tough materials, but torsion is a relatively neglected loading mode. The wood skeletons of cholla cacti, subject to spartan desert conditions and hurricane force winds, provide a new template for torsionally resilient biological materials. Novel mesostructural characterization methods of laser-scanning and photogrammetry are used alongside traditional optical microscopy, scanning electron microscopy, and micro-computed tomography to identify mechanisms responsible for torsional resistance. These methods, in combination with finite element analysis reveal how cholla meso and macro-porosity and fibril orientation contribute to highly density-efficient mechanical behavior. Selective lignification and macroscopic tubercle pore geometry contribute to density-efficient shear stiffness, while mesoscopic wood fiber straightening, delamination, pore collapse, and fiber pullout provide extrinsic toughening mechanisms. These energy absorbing mechanisms are enabled by the hydrated material level properties. Together, these hierarchical behaviors allow the cholla to far exceed bamboo and trabecular bone in its ability to combine specific torsional stiffness, strength, and toughness. Statement of Significance The Cholla cactus experiences, due to the high velocity desert winds, high torsional loads. Our study has revealed the amazingly ingenious strategy by which the tubular structure containing arrays of voids intermeshed with wood fibers resists these high loads. Deformation is governed by compressive and tensile stresses which are greatest at 45 degrees to the cross section. It proceeds by stretching, sliding, and bending of the wood fibers which are coupled with the pore collapse, resulting in delayed failure and a high torsional toughness.

The goal of the present textbook is to introduce the basics of the finite element analysis for thermal and deformation problems that can be analyzed with linear models. In this book we focus on the following aspects of which we believe the user of finite element analysis must have a basic understanding: 1. Understanding of the mathematical models: what are the underlying assumptions, how is the model properly defined, what are the solutions of elementary models. 2. How is the finite element model related to the mathematical model: what is the effect of the conversion of the continuous mathematical model to the discrete finite element model, what is the accuracy, how to control the accuracy. 3. How to use results from finite element modeling: interpretation of results, estimation of errors, verification and validation, proper formulation of reports. In this respect the present textbook differs from the current popular engineering textbooks on finite elements. The mathematics is also presented as simply as possible, so that the present textbook is also quite different from state of the art the mathematical treatments of the finite element method.

This paper investigates a method for improving the accuracy of the stress predicted from models using the mean-strain finite elements recently proposed by Krysl and collaborators [IJNME 2016, 2017]. In state-of-the-art finite element programs, the stress values at the integration points are commonly post-processed to obtain nodal values of stress. The mean stresses are element-wise constant, and hence the nodal values obtained from the mean stresses tend to be of lower accuracy. The proposed method post-processes the uniform stress in each element in combination with a linearly-varying stabilization stress field to produce a more accurate representation of the nodal stresses. Selected examples are presented to demonstrate improvements achievable with the proposed methodology for hexahedral and quadratic tetrahedral mean-strain finite elements.

A broad-ranging plasticity model is constructed by combining Bigoni-Piccolroaz yield function with a sweeping expression of kinematic hardening. Subsequently, two explicit and implicit update algorithms are developed to compute the stress from the rate equations of the plasticity in an incremental elastoplastic finite element analysis. The integrations are based on Forward and Backward Euler schemes. Considering a number of varying materials and the associated plasticity models, the proposed formulations are examined by means of a comprehensive set of numerical investigations. These include investigations to verify the suggested strategies along with two numerical simulations carried out by implementing the algorithms in a nonlinear finite element code. The versatility of the formulae allows for an expansive choice of elastoplastic constitutive models with a simple adjustment of a few parameters.

A simple, lightweight, and flexible, package in the programming language Julia for managing finite element mesh data structures is presented. The key role in the design of the data structures is granted to the incidence relation. This concept has some interesting implications for the simplicity and efficiency of the implementation. The entire library has less than 500 executable lines. The low memory requirements are also notable. The design of the data structures is not fixed a priori. The user of the library is given the power over the decisions which mesh entities should be represented explicitly in the data structures, and which of the topological relationships should be computed and stored. This enables a small memory footprint, yet affords a sufficiently rich topology description capability.

Summary A mean-strain 10-node tetrahedral element is developed for the solution of geometrically nonlinear solid mechanics problems using the concept of energy-sampling stabilization. A uniform-strain tetrahedron for applications to linear elasticity was recently described. The formulation as extended here is able to solve large-strain hyperelasticity. The present 10-node tetrahedron is composed of several four-node linear tetrahedral elements, four tetrahedra in the corners, and four tetrahedra that tile the central octahedron in three possible sets of four-node tetrahedra, corresponding to three different choices for the internal diagonal. We formulate a mean-strain element with stabilization energy evaluated on the four corner tetrahedra, which is shown to guarantee consistency and stability. The stabilization energy is expressed through a stored-energy function, and contact with input parameters in the small-strain regime is made. The neo-Hookean model is used to formulate the stabilization energy. As for small-strain elasticity, the stabilization parameters are determined by actual material properties and geometry of a tetrahedra without any user input. The numerical tests demonstrate that the present element performs well for solid, shell, and nearly incompressible structures. Copyright © 2016 John Wiley & Sons, Ltd.

Modern bony fishes display a high morphological diversity in their auditory structures. Yet, unraveling the sound-induced in situ interaction of the auditory structures has been challenging. Synchrotron radiation-based imaging techniques with high spatial and temporal resolutions now provide a powerful tool to study the functional morphology of such structures in a non-invasive way. We investigated two species representing two types of otophysic connections, namely, goldfish (Weberian apparatus) and Etroplus canarensis (swimbladder extensions contacting the inner ears) asking how different ancillary auditory structures affect otolith motion. Fishes were subjected to a 200 Hz pure tone in a 2L-standing wave tube-like setup while performing 2D radiography based on hard x-ray phase-contrast imaging. The shakers at each end of the tube were either driven 0° in-phase or 180° out-of-phase resulting in maximized sound pressure or sound-induced particle motion in the tube center, respectively. In both species, saccular otolith motion was more pronounced when the swimbladder walls oscillated under 0° in-phase condition and the motion patterns mainly matched the respective orientation patterns of ciliary bundles on the sensory epithelia. In future, we will quantify the motion patterns of fish auditory structures by applying “sound tomography” to cover the three-dimensional aspect of the moving structures.

A method for stabilizing the mean-strain hexahedron was described by Krysl (in IJNME 2014). The technique relied on a sampling of the stabilization energy using the mean-strain quadrature and the full Gaussian integration rule, which was shown to guarantee consistency and stability. The stabilization energy was assumed to be generated by a modified constitutive matrix based on the spectral decomposition. The stabilization required user-selected values of the stabilization parameters. In the present work we eliminate the arbitrariness of the stabilization parameters. We formulate the technique more precisely as an assumed-strain method, and we express the stabilization energy in terms of input parameters of the real material. Finally, we fix the value of the stabilization parameters in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element. For simplicity the developments are limited to linear elasticity, but with an arbitrarily anisotropic elasticity matrix. The accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements as implemented in Abaqus and with a number of successful hexahedral and shell elements and we demonstrate that the present element performs very well when used with large aspect ratios for thin structures such as plates or shells.

The initial boundary value problem of the transverse vibration of a taut string is a classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic bar also serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalent in that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbach of 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.

Following publication of the original article [1], the authors reported the errors in the equations.

This work presents an efficient way to calculate the added mass matrix, which allows solving for natural frequencies and modes of solids vibrating in an inviscid and infinite fluid. The finite element method (FEM) is used to compute the vibration spectrum of a dry structure, then the boundary element method (BEM) is applied to compute the pressure modes needed to determine the added mass matrix that represents the fluid. The BEM requires numerical integration which results in a large computational cost. In this work, a reduction of the computational cost was achieved by computing the values of the pressure modes with the required numerical integration using a coarse BEM mesh, and then, interpolation was used to compute the pressure modes at the nodes of a fine FEM mesh. The added mass matrix was then computed and added to the original mass matrix of the generalized eigenvalue problem to determine the wetted natural frequencies. Computational cost was minimized using a reduced eigenvalue problem of size equal to the requested number of natural frequencies. The results show that the error of the natural frequencies using the procedure in this work is between 2% and 5% with 87% reduction of the computational time. The motivation of this work is to study the vibration of marine mammals' ear bones.

Nodally integrated elements exhibit spurious modes in dynamic analyses (such as in modal analysis). Previously published methods involved a heuristic stabilization factor, which may not work for a large range of problems, and a uniform amount of stabilization was used over all the finite elements in the mesh. The method proposed here makes use of energy-sampling stabilization. The stabilization factor depends on the shape of the element and appears in the definition of the properties of a stabilization material. The stabilization factor is non-uniform over the mesh, and can be computed to alleviate shear locking, which directly depends on the aspect ratios of the finite elements. The nodal stabilization factor is then computed by volumetric averaging of the element-based stabilization factors. Energy-sampling stabilized nodally integrated elements (ESNICE) tetrahedral and hexahedral are proposed. We demonstrate on examples that the proposed procedure effectively removes spurious (unphysical) modes both at lower and at higher ends of the frequency spectrum. The examples shown demonstrate the reliability of energy-sampling in stabilizing the nodally integrated finite elements in vibration problems, just sufficient to eliminate the spurious modes while imparting minimal excessive stiffness to the structure. We also show by the numerical inf-sup test that the formulation is coercive and locking-free.

Summary Finite element analysis on CT-scanned cetacean heads can be used to determine the path of sounds through the head. These cetacean sound paths are different from those of land mammals, and, surprisingly, they also differ depending on the frequency of the sound. Using these, it was possible to model audiogram for mysticetes and compare the methods of hearing between cetacean taxa.

The fluid-structure interaction technique provides a paradigm for solving scattering from elastic structures embedded in an environment characterized by a Green's function, by a combination of finite and boundary element methods. In this technique, the finite element method is used to discretize the equations of motion for the structure and the Helmholtz-Kirchhoff integral with the appropriate Green's function is used to produce the discrete pressure field in the exterior medium. The two systems of equations are coupled at the surface of the structure by imposing the continuity of pressure and normal particle velocity. The present method condenses the finite element model so that finally only the boundary element problem needs to be solved. This results in a significant reduction in the number of unknowns and hence a much lower cost. In this paper, the fluid-structure interaction method is specialized to axially-symmetric objects for non-axially-symmetric loading in free space using a circumferential Fourier expansion of the fields. The specialization of the method to axially-symmetric objects results in even further significant reductions in computation. The method is validated using well-known benchmark solutions. A derivation of the method for an arbitrarily-shaped elastic structure embedded in an arbitrary environment characterized by a Green's function is given in the Appendix.

Summary A method for stabilizing the mean-strain hexahedron for applications to anisotropic elasticity was described by Krysl (2015). The technique relied on a sampling of the stabilization energy using the mean-strain quadrature and the full Gaussian integration rule. This combination was shown to guarantee consistency and stability. The stabilization energy was expressed in terms of input parameters of the real material, and the value of the stabilization parameter was fixed in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element (Krysl, submitted). Here, the formulation is extended to large-strain hyperelasticity (as an example, the formulation allows for inelastic behavior to be modeled). The stabilization energy is expressed through a stored-energy function, and contact with input parameters in the small-strain regime is made. As for small-strain elasticity, the stabilization parameter is determined to optimize bending performance. The accuracy and convergence characteristics of the present formulations for both solid and thin-walled structures (shells) compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements described in the open literature and also with a number of successful shell elements. Copyright © 2015 John Wiley & Sons, Ltd.

SUMMARY In this work, a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems. The element stems from the Nodally Integrated Continuum Element (NICE) formulation and its extensions. Assumed gradient operators are derived via nodal integration from the kinematic-weighted residual; the degrees of freedom are only the displacements at the nodes. The adopted constitutive model is the classical associative von Mises plasticity model with isotropic and kinematic hardening; in particular, a double-step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations. Efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.

Summary A method for stabilizing the mean-strain hexahedron is described that differs from the currently known approaches. For simplicity, the developments are limited to linear elasticity but with an arbitrarily anisotropic elasticity matrix. The technique relies on a sampling of the stabilization energy using two quadrature rules, the mean-strain quadrature and the full Gaussian integration rule. The use of two quadrature rules is shown to guarantee consistency and stability. The stabilization energy is assumed to be generated by a modified constitutive matrix based on the spectral decomposition. The spectral decomposition of the constitutive matrix identifies the stiff and flexible modes of deformation. The stiff modes of deformation are only sampled by the mean-strain integration, which eliminates volumetric locking for isotropic materials as well as locking due to strongly anisotropic material properties. The accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements as implemented in ABAQUS. Copyright © 2014 John Wiley & Sons, Ltd.

SUMMARY Anisotropic elastic materials, such as the homogenized model of a fiber-reinforced matrix, can display near rigidity under certain applied stress–the resulting strains are small compared with the strains that would occur for other stresses of comparable magnitude. The anisotropic material could be rigid under hydrostatic pressure if the material were incompressible, as in isotropic elasticity, but also for other stresses. Some commonly used finite elements are effective in dealing with incompressibility, but are ill-equipped to handle materials that lock under non-hydrostatic stress states (e.g., uniformly reduced serendipity and Q1/Q0 B-bar hexahedra). The failure of the original B-bar method is attributed to the assumption that the mode of deformation to be relieved is one of near incompressibility. The remedy proposed here is based on the spectral decomposition of the compliance matrix of the material. The spectrum can be interpreted to separate nearly-rigid and flexible modes of stress and strain, which leads naturally to a generalized selective reduced integration. Furthermore, the spectral decomposition also enables a three-field elasticity formulation that results in a B-bar method that is effective for general anisotropic materials with an arbitrary nearly-rigid mode of deformation.Copyright © 2014 John Wiley & Sons, Ltd.

Psychoacoustic laboratory studies with live dolphins require considerable resources and are essential for assessing the validity of our models. Computerized numerical modelling methods are a reasonable approach to simulate the vibroacoustic functions of the dolphin biosonar apparatus. In order to validate this approach, we chose a vibroacoustic finite element model to simulate sound production and sound beam formation in the bottlenose dolphin (Tursiops truncatus), based on computed tomography scans from live and postmortem dolphins. The right and left dorsal bursae were assumed to be potential sound sources. The simulations confirm several hypotheses: (1) the shape of the skull plays a role in the formation of the sound transmission beam; (2) the melon appears to concentrate the acoustic energy by a factor of four in the transmitted beam; (3) focusing the sound beam apparently occurs in a series of stages that include contributions from the skull, nasal diverticula, melon and connective tissue structures...

Odontocete ear complexes or tympanoperiotic complexes (TPCs) were compared for asymmetry. Left and right TPCs were collected from one long-beaked common dolphin (Delphinus capensis) and one Amazon River dolphin (Inia geoffrensis). Asymmetry was assessed by volumetric comparisons of left and right TPCs and by visual comparison of superimposed models of the right TPC to a reflected mirror image of the left TPC. Kolmogorov–Smirnov tests were performed to compare the resonant frequencies of the TPCs as calculated by vibrational analysis. All analyses found slight differences between TPCs from the same specimen in contrast to the directional asymmetry in the nasal region of odontocete skulls.

The head-related transfer function (HRTF) is an important descriptor of spatial sound field reception by the listener. In this study, we computed the HRTF of the common dolphin Delphinus delphis. The received sound pressure level at various locations within the acoustic fats of the internal pinna near the surface of the tympanoperiotic complex (TPC) was calculated for planar incident waves directed toward the animal. The relative amplitude of the received pressure versus the incident pressure was the representation of the HRTF from the point of view of the animal. It is of interest that (1) different locations on the surface of the TPC resulted in different HRTFs, (2) the HRTFs for the left and right ears were slightly asymmetric, and (3) the locations of the peaks of the HRTF depended on the frequency of the incident wave.

In 1974, Norris and Harvey published an experimental study of sound transmission into the head of the bottlenose dolphin. We used this rare source of data to validate our Vibroacoustic Toolkit, an array of numerical modeling simulation tools. Norris and Harvey provided measurements of received sound pressure in various locations within the dolphin’s head from a sound source that was moved around the outside of the head. Our toolkit was used to predict the curves of pressure with the best-guess input data (material properties, transducer and hydrophone locations, and geometry of the animal’s head). In addition, we performed a series of sensitivity analyses (SAs). SA is concerned with understanding how input changes to the model influence the outputs. SA can enhance understanding of a complex model by finding and analyzing unexpected model behavior, discriminating which inputs have a dominant effect on particular outputs, exploring how inputs combine to affect outputs, and gaining insight as to what additional information improves the model’s ability to predict. Even when a computational model does not adequately reproduce the behavior of a physical system, its sensitivities may be useful for developing inferences about key features of the physical system. Our findings may become a valuable source of information for modeling the interactions between sound and anatomy.

The nodally integrated plate element formulation is an assumed strain finite element technique for shear deformable plates that enforce weakly the balance and the kinematic equations. The strain-displacement operators are derived via nodal integration satisfying a priori the kinematic weighted residual statement. The present work analyzes the NIPE technique, including the new element, by testing thoughtfully the sensitivity of the elements to severe geometry distortions. We propose an improvement that confers robustness to all element shapes developed by the NIPE formulation. We present also a new nine-node NIP element configuration. The new nine-node element uses bi-quadratic interpolations of the transverse displacement and rotations and is computed by means of a nine-node quadrature rule. A brief review of the improved triangular and quadrangular NIPEs is reported for elastic plate analyses. A few challenging benchmarks carried out on extreme distorted meshes illustrate the performance of the introduced NIP-Q9 element. We detail that the new NIPE formulation confers insensitivity to extreme distortions for the quadratic quadrilateral element and allows to solve for severe distortion with the NIPE family.

SUMMARY A finite element model for linear-elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain-displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four-node and linear hexahedral eight-node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf-sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.