Nonlinear Displacement Field Research Articles

HYMALAIA: a hybrid lagrangian model for intrinsic alignments

ABSTRACT The intrinsic alignment of galaxies is an important ingredient for modelling weak-lensing measurements, and a potentially valuable cosmological and astrophysical signal. In this paper, we present HYbrid Model Advected from LAgrangian space for IA (HYMALAIA): a new model to predict the intrinsic alignments of biased tracers. HYMALAIA is based on a perturbative expansion of the statistics of the Lagrangian shapes of objects, which is then advected to Eulerian space using the fully non-linear displacement field obtained from N-body simulations. We demonstrate that HYMALAIA is capable of consistently describing monopole and quadrupole of halo shape–shape and matter–shape correlators, and that, without increasing the number of free parameters, it does so more accurately than other perturbatively inspired models such as the non-linear alignment model and the tidal-alignment-tidal-torquing model.

Numerical model of nonlinear elastic bulk wave propagation in solids for non-destructive evaluation

Nonlinear ultrasonic techniques can be difficult and non-intuitive to understand due to the range of wave mixing combinations available between similar or different wave modes. To overcome this, a numerical model that uses a Finite Difference Time Domain (FDTD) scheme, without a staggered grid system, to solve the nonlinear elastic bulk wave equations in two dimensions is proposed in this paper, with the purpose of better understanding nonlinear ultrasonic techniques. Both material and geometrical nonlinearities are considered and a stress-type boundary condition is used to model the excitation. The energy transfer between frequency bands is shown and the amplitude trend of the harmonics are validated against available theories. It is then used to simulate the nonlinear ultrasonic field generated by a finite-size element of an array in a solid to better understand its behaviour. An array-element directivity at the second harmonic frequency is shown. Since the FDTD model does not account for attenuation, a correction using a ray-based approach is applied to the simulated result for direct comparison against experimental measurements. The field obtained from a typical array used in non-destructive testing is then simulated to characterise its behaviour. Experimental comparison of the nonlinear displacement fields for different focal positions showed good agreement, further validating the FDTD model. This FDTD model opens opportunities to virtually experiment and design appropriate nonlinear ultrasonic array inspections whilst isolating and understanding contribution from material nonlinearity.

Efficient zigzag theory-based spectral element model for guided waves in composite structures containing delaminations

Lamb wave-based structural health monitoring offers excellent potential for real-time delamination detection in laminated structures. It necessitates fast and accurate numerical simulation of guided wave interaction with delaminations. Towards this objective, we present an efficient layerwise zigzag theory (ZIGT) based time-domain spectral element for wave propagation analysis of composite beam- and panel-type structures (strips) containing delamination at arbitrary locations. The ZIGT delivers accuracy like layerwise theories by allowing slope discontinuities in the in-plane displacement field at layer interfaces while maintaining computational efficiency like equivalent single-layer (ESL) theories, making it ideal for such analyses. The high-order elemental nodes at Lobatto points have only four displacement variables, irrespective of the number of layers in the laminate. Delamination is modelled using the region approach, adapting the recently proposed hybrid point-least squares continuity method to satisfy the nonlinear displacement field continuity at the delamination fronts. The model’s efficacy is examined vis-à-vis elasticity-based elements, ESL theory-based elements, and the ZIGT-based standard finite element for free vibration and guided wave propagation responses of composite strips featuring multiple delaminations. The present model shows superior overall efficiency, accuracy, and convergence. An application of the model is also explored for identifying delamination locations from Lamb wave velocity fields.

Lagrangian displacement field estimators in cosmology

The late-time nonlinear Lagrangian displacement field is highly correlated with the initial field, so reconstructing it could enable us to extract primordial cosmological information. Our previous work [A. Ota et al., Phys. Rev. D 104, 123508 (2021)] carefully studied the displacement field reconstructed from the late-time density field using the iterative method proposed by Schmittfull et al. [Phys. Rev. D 96, 023505 (2017)] and found that it does not fully converge to the true, underlying displacement field (e.g., $\ensuremath{\sim}8%$ offset at $k\ensuremath{\sim}0.2\text{ }\text{ }h\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ at $z=0.6$). We also constructed the Lagrangian perturbation theory model for the reconstructed field, but the model could not explain the discrepancy between the true and the reconstructed fields in the previous work. The main sources of the discrepancy were speculated to be a numerical artifact in the displacement estimator due to the discreteness of the sample. In this paper, we develop two new estimators of the displacement fields to reduce such a numerical discreteness effect, the normalized momentum estimator and the rescaled resumed estimator. We show that the discrepancy Ota et al. [Phys. Rev. D 104, 123508 (2021)] reported is not due to the numerical artifacts. We conclude that the method from Schmittfull et al. [Phys. Rev. D 96, 023505 (2017)] cannot fully reconstruct the shape of the nonlinear displacement field at the redshift we studied, while it is still an efficient baryon acoustic oscillation reconstruction method. In parallel, by properly accounting for the UV-sensitive term in a reconstruction procedure with an effective field theory approach, we improve the theoretical model for the reconstructed displacement field, by almost 5 times, from $\ensuremath{\sim}15%$ to the level of a few percent at $k\ensuremath{\sim}0.2\text{ }\text{ }h\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ at the redshift $z=0.6$.

SenseNet: A Physics-Informed Deep Learning Model for Shape Sensing

Shape sensing is an emerging technique for the reconstruction of deformed shapes using data from a discrete network of strain sensors. The prominence is due to its suitability in promising applications such as structural health monitoring in multiple engineering fields and shape capturing in the medical field. In this work, a physics-informed deep learning model, named SenseNet, was developed for shape sensing applications. Unlike existing neural network approaches for shape sensing, SenseNet incorporates the knowledge of the physics of the problem, so its performance does not rely on the choices of the training data. Compared with numerical physics-based approaches, SenseNet is a mesh-free method, and therefore it offers convenience to problems with complex geometries. SenseNet is composed of two parts: a neural network to predict displacements at the given input coordinates, and a physics part to compute the loss using a function incorporated with physics information. The prior knowledge considered in the loss function includes the boundary conditions and physics relations such as the strain–displacement relation, material constitutive equation, and the governing equation obtained from the law of balance of linear momentum. SenseNet was validated with finite-element solutions for cases with nonlinear displacement fields and stress fields using bending and fixed tension tests, respectively, in both two and three dimensions. A study of the sensor density effects illustrated the fact that the accuracy of the model can be improved using a larger amount of strain data. Because general three dimensional governing equations are incorporated in the model, it was found that SenseNet is capable of reconstructing deformations in volumes with reasonable accuracy using just the surface strain data. Hence, unlike most existing models, SenseNet is not specialized for certain types of elements, and can be extended universally for even thick-body applications.

Numerical Model for a Geometrically Nonlinear Analysis of Beams with Composite Cross-Sections

This paper presents a beam model for a geometrically nonlinear stability analysis of the composite beam-type structures. Each wall of the cross-section can be modeled with a different material. The nonlinear incremental procedure is based on an updated Lagrangian formulation where in each increment, the equilibrium equations are derived from the virtual work principle. The beam model accounts for the restrained warping and large rotation effects by including the nonlinear displacement field of the composite cross-section. First-order shear deformation theories for torsion and bending are included in the model through Timoshenko’s bending theory and a modified Vlasov’s torsion theory. The shear deformation coupling effects are included in the model using the six shear correction factors. The accuracy and reliability of the proposed numerical model are verified through a comparison of the shear-rigid and shear-deformable beam models in buckling problems. The obtained results indicated the importance of including the shear deformation effects at shorter beams and columns in which the difference that occurs is more than 10 percent.

Stability analysis of shear deformable cross-ply laminated composite beam-type structures

This paper presents a shear deformable beam model for the nonlinear stability analysis of composite beam-type structures. Each wall of the cross-section is stacked in the cross-ply scheme. The incremental equilibrium equations are derived in the framework of an updated Lagrangian formulation. Hooke’s law is assumed to be valid. The nonlinear displacement field of the composite cross-section accounts for the restrained warping and the large rotation effects. The shear deformation effects are included through Timoshenko’s bending theory and modified Vlasov’s torsion theory. Additional shear correction factors due to the bending-torsion coupling occurring at asymmetric cross-sections are introduced in the analysis. The accuracy and reliability of proposed numerical mode are verified on benchmark examples, and the obtained results confirm that it can be classified as a shear locking-free model.

Nonlinear deformation monitoring of elastic beams based on isogeometric iFEM approach

Shape sensing plays a key role in Structural Health Monitoring (SHM) and has become an excellent methodology for large-scale engineering structures to achieve significant improvement in their safety, reliability, and affordability. The inverse finite element method (iFEM) is an accurate and efficient method for shape sensing to reconstruct the three-dimensional displacements using in situ surface strain data. This study proposes a novel shape sensing method for large deformation monitoring based on strain gradient theory and iFEM method. Initially, the nonlinear displacement fields are described, and Green–Lagrange strain theory is employed to deduce the theoretical section strains, and then the strain–displacement relation is established by using a least-squares variational principle. Next, isogeometric displacement functions and the measured section strain formulations are deduced, and a decoupling method for high order section strains is proposed. Finally, a cantilever beam is used to demonstrate the accuracy and effectiveness of the refined isogeometric iFEM method for large deformations. The numerical results show the superior reconstruction capability and potential applicability of the proposed model for accurate shape prediction of the non-linear deformation of beam structure.

Physics informed neural networks for continuum micromechanics

Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. The principle idea is the usage of a neural network as a global ansatz function for partial differential equations. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong nonlinear solution fields by optimization. In this work we consider nonlinear stress and displacement fields invoked by material inhomogeneities with sharp phase interfaces. This constitutes a challenging problem for a method relying on a global ansatz. To overcome convergence issues, adaptive training strategies and domain decomposition are studied. It is shown, that the domain decomposition approach is capable to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world μCT-scans.

Modeling iterative reconstruction and displacement field in the large scale structure

The next generation of galaxy surveys like the Dark Energy Spectroscopic Instrument (DESI) and Euclid will provide datasets orders of magnitude larger than anything available to date. Our ability to model nonlinear effects in late time matter perturbations will be a key to unlock the full potential of these datasets, and the area of initial condition reconstruction is attracting growing attention. Iterative reconstruction developed in Ref. [1] is a technique designed to reconstruct the displacement field from the observed galaxy distribution. The nonlinear displacement field and initial linear density field are highly correlated. Therefore, reconstructing the nonlinear displacement field enables us to extract the primordial cosmological information better than from the late time density field at the level of the two-point statistics. This paper will test to what extent the iterative reconstruction can recover the true displacement field and construct a perturbation theory model for the postreconstructed field. We model the iterative reconstruction process with Lagrangian perturbation theory~(LPT) up to third order for dark matter in real space and compare it with $N$-body simulations. We find that the simulated iterative reconstruction does not converge to the nonlinear displacement field, and the discrepancy mainly appears in the shift term, i.e., the term correlated directly with the linear density field. On the contrary, our 3LPT model predicts that the iterative reconstruction should converge to the nonlinear displacement field. We discuss the sources of discrepancy, including numerical noise/artifacts on small scales, and present an ad hoc phenomenological model that improves the agreement.

On large deformation and stability of microcantilevers under follower load

In the present study, a novel geometrically exact nonlinear beam model based on the nonlinear displacement field, centerline-inextensibility condition, Euler-Bernoulli beam hypothesis, and modified couple stress theory is established to investigate the extremely large deformations and stability characteristics of size-dependent microcantilevers subjected to a tip-concentrated non-conservative (follower type) load with arbitrary inclination angle. The newly developed mathematical formulation explains the size effect phenomenon in the microcantilevers with only one material length scale parameter. The nonlinear shooting method for two-point boundary value problems is utilized to assess the deformed configuration of microsystem based on the quasistatic form of nonlinear mathematical model. Afterwards, the linearized dynamical mathematical model around the nonlinear deformed configurations of microsystem is numerically solved using the Galerkin technique to examine the stability of obtained responses. A comprehensive investigation is then conducted to highlight the role of size-dependency in the critical follower force at which the microsystem undergoes a flutter instability, as well as the nonlinear large deformation of microsystem in the pre-flutter region.

LOW-VELOCITY IMPACT RESPONSES OF FG-GRC SANDWICH BEAMS WITH VISCOELASTIC CORES

The dynamic snap-through and oscillation behaviors are studied of a post-buckled functionally graded graphene reinforced composite (FG-GRC) beam with a viscoelastic core under low-velocity impact. The modified Halpin Tsai meso-mechanical model is used to predict the material properties of FG-GRC. The Hertz point contact model is used to calculate the contact force between the impactor and the post-buckled beam. Considering the axial prestress, the constitutive relations of the composite layer and the Kelvin type viscoelastic constitutive model of the damping layer are proposed. A generalized higher-order shear deformation zig-zag beam theory is utilized to model the nonlinear displacement fields. Based on the Hamilton energy variational principle, the governing equations of dynamics are derived. Through two-step analysis, the post-buckling equilibrium paths are obtained as the initial state condition for the impact problem analysis. Further, combined with the fourth-order Runge Kutta method, the two-step perturbation-Galerkin method is extended to simulate the time history curves of the contact force and the dynamic response curves of the post-buckled beams. Compared with the results based on other beam models, the correctness of the material model, theoretical model and computation method is validated. The dynamic characteristics of bi-stable large amplitude vibration of the post-buckled beams under single and two collisions are studied. The effects of axial load, impact velocity, viscoelastic damping characteristics and impactor materials on the contact force and the deflection time history curves of the post-buckled beams are discussed. The results suggest that the contact force is only sensitive to the impact velocity. A certain structural collision parameter design can change the response of the post-buckled beam from single potential energy trap motion to double trap large oscillation, and the contact force is approximately unchanged. The results of this work can be of reference significance for the design of bi-stable energy capture system with collision.

A coupled efficient layerwise finite element model for free vibration analysis of smart piezo-bonded laminated shells featuring delaminations and transducer debonding

This paper presents a computationally efficient multifield finite element (FE) model for accurate analysis of smart composite and sandwich shells equipped with piezoelectric patch actuators and sensors, featuring multiple delaminations and transducer debonding. It employs a facet shell element with four physical nodes and one electric node, based on the third order zigzag theory approximations for displacements and a piecewise quadratic through-thickness variation for the electric potential. A hybrid point-least squares method recently proposed by the authors is extended to impose the condition of continuity of the nonlinear displacement field at the delamination, debonding, and patch fronts. The methodology is generic for multiple patch transducers, delaminations, and debonding, occurring at any arbitrary interface and in-plane locations. The model shows excellent accuracy in comparison with the coupled full-field 3D FE solutions, for the natural frequencies and complex global, local and hybrid mode shapes of delaminated smart composite shells and soft-core sandwich plates, with partially debonded actuators and sensors. The developed model will be of immense use for the design of active vibration control and structural health monitoring strategies for smart laminated shell-type structures featuring such damages.

Modeling of 3-D elastic waves in nonlinear elastic solids using the iterative nonlinear contrast source method

In nondestructive testing, the generation of higher harmonics and the mixing of elastic waves are used to measure the nonlinearity parameters, which in turn are closely related to the microscopic state of the material. A 3-D numerical tool that can simulate large scale, four-dimensional, nonlinear elastic wavefields would be very useful for the reliable interpretation of experimental results. Here, a method is presented that can efficiently compute the nonlinear displacement fields in a homogeneous, isotropic elastic medium, taking into account its third-order elastic constants (A, B, C). The method is based on the Neumann iterative solution of an integral equation involving a Green's function of the linear `background' medium, and a contrast source representing the nonlinearity of the medium. The integral equation is solved iteratively, and the computations are based on Fast Fourier Transforms using a sampling rate close to the Nyquist limit, i.e., two grid points of the shortest wavelength. The displacement fields are evaluated using the scalar and vector potential functions representing the compressional and shear displacements. In this presentation, the suitability of the INCS method for modeling the harmonics of 3-D nonlinear elastic waves and its validation by comparison with an analytical benchmark solution, will be presented.

Comparison of classical and refined beam models applied on isotropic and FG thin-walled beams in nonlinear buckling response

In this work, nonlinear buckling analysis of thin-walled beams by two different models is investigated: first one, developed by MUL2 group, based on Carrera Unified Formulation, and THINWALL v.16 beam model based on classical Euler-Bernoulli-Vlasov beam theory. In former, the refined beam theories are obtained on the basis of Taylor and Lagrange-type expansions. Latter is performed in framework of updated Lagrangian formulation adopting nonlinear displacement field of thin-walled cross-section. Isotropic and FG beams are referred to. It is observed that numerical results obtained by these methods match very well.

Delamination modeling in doubly curved laminated shells for free vibration analysis using zigzag theory‐based facet shell element and hybrid continuity method

SummaryWe present a finite element (FE) formulation for the free vibration analysis of doubly curved laminated composite and sandwich shells having multiple delaminations, employing a facet shell element based on the efficient third‐order zigzag theory and the region approach of modeling delaminations. The methodology, hitherto not attempted, is general for delaminations occurring at multiple interfacial and spatial locations. A recently developed hybrid method is used for satisfying the continuity of the nonlinear layerwise displacement field at the delamination fronts. The formulation is shown to yield very accurate results with reference to full‐field three‐dimensional FE solutions, for the natural frequencies and mode shapes of delaminated shallow and deep, composite and highly inhomogeneous soft‐core sandwich shells of different geometries and boundary conditions, with a significant computational advantage. The accuracy is sensitive to the continuity method used at the delamination fronts, the usual point continuity method yielding rather poor accuracy, and the proposed hybrid method giving the best accuracy. Such efficient modeling of laminated shells with delamination damage will be of immense use for model‐based techniques for structural health monitoring of laminated shell‐type structures.

Second order shear deformation theory for functionally graded axisymmetric thick shell with variable thickness under non-uniform pressure

This paper studies Second order Shear Deformation Theory (SSDT), as a higher order shear deformation theory, for an axisymmetric functionally graded shell of revolution with variable thickness. According to symmetrical condition, there is not any displacement and any variation along the symmetric direction. The governing equations of proposed shell are derived by virtual work principle. As a special case, the governing equations are rewritten for a cylinder with variable thickness and non-uniform internal pressure. The comparison between current work, classical theory and First order Shear Deformation Theory (FSDT) are illustrated with some numerical results. Non-homogenous material, boundary condition effects, non-uniform pressure, arbitrary curvature with variable thickness and nonlinear displacement field are some advantages of current work. The optimum design of shell is the main goal of current approach that has a wide industrial application like aerospace engineering.

ЛАГРАНЖЕВ ФОРМАЛИЗМ В ЗАДАЧАХ О МАЛЫХ КОЛЕБАНИЯХ ВИХРЕВЫХ ТЕЧЕНИЙ И ЕГО СВЯЗЬ С ВАРИАЦИОННЫМ ПРИНЦИПОМ ДЛЯ ИДЕАЛЬНОЙ НЕСЖИМАЕМОЙ ГИДРОДИНАМИКИ ВИХРЕВЫХ НИТЕЙ

The paper discusses the description of vortex flows of an ideal incompressible fluid based on the formalism of Lagrangian mechanics. Using the displacement field of liquid particles as a generalized coordinate, we write out the Lagrangian describing the dynamics of small perturbations (Kopiev, Chernyshev, 2018). The corresponding Lagrange equations are the equation for the displacement field (Drazim, Reid, 1981): This equation is equivalent to the Helmholtz equation for vorticity perturbations. The displacement field is defined as the difference in the positions of liquid particles on trajectories in disturbed and undisturbed flows. Although this definition is given in terms of Lagrangian variables associated with liquid particles, the displacement field itself is an Euler variable, expressed through velocity and vorticity perturbations. An example of using Lagrangian to solve the problem of conservation of the quadrupole moment of a vortex flow is considered. Using the Noether theorem, conditions on a stationary flow are obtained, under which the quadrupole moment of small perturbations of this flow is an integral of motion (Kopiev, Chernyshev, 2018). It is shown that these conditions are satisfied for the jet flows uniform along the longitudinal coordinate. The result obtained is important in aeroacoustics due to the fact that the quadrupole moment of the vortex flow represents the main term of the decomposition of a compact acoustic source in Machnumber (Lighthill, 1952; Crow, 1970; Kopiev, Chernyshev, 1995). The generalization of these results to the nonlinear case is considered. The Lagrangian is obtained for an arbitrary nonlinear displacement field: nowhere Gis Green’s function of the Laplace equation. The corresponding Lagrange equations coincide with the differential equations describing the nonlinear dynamics of the displacement field (Drazin, Reid, 1981). Expansion of the Lagrangian in small perturbations to quadratic terms gives the Lagrangian of the linear system. The question of the relationship of the proposed approach to the description of the dynamics of an incompressible fluid and known approaches based on the formalism of Lagrangian mechanics with the coordinates of liquid particles as generalized coordinates (Chapman, 1978; Goncharov, Pavlov, 2008; Kuznetsov, Ruban, 1998) is considered. It is shown that the transformation of the Lagrangian obtained in (Kuznetsov, Ruban, 1998) to the Lagrangian can be carried out by transforming Lagrangian variables (coordinates of liquid particles) to Eulerian variables (displacement field). This study was supported by the Russian Science Foundation, project No. 17-11-01271.

Kirigami Mechanics as Stress Relief by Elastic Charges.

We develop a geometric approach to understand the mechanics of perforated thin elastic sheets, using the method of strain-dependent image elastic charges. This technique recognizes the buckling response of a hole under an external load as a geometrically tuned mechanism of stress relief. We use a diagonally pulled square paper frame as a model system to quantitatively test and validate our approach. Specifically, we compare nonlinear force-extension curves and global displacement fields in theory and experiment. We find a strong softening of the force response accompanied by curvature localization at the inner corners of the buckled frame. Counterintuitively, though in complete agreement with our theory, for a range of intermediate hole sizes, wider frames are found to buckle more easily than narrower ones. Upon extending these ideas to many holes, we demonstrate that interacting elastic image charges can provide a useful kirigami design principle to selectively relax stresses in elastic materials.

Digital Image Correlation Analysis of Displacement Based on Corrected Three Surface Fitting Algorithm

SFA (Surface Fitting Algorithm) for continuous displacement is an important method for digital image correlation with antinoise ability and computational efficiency advantages in practical applications. In order to improve the algorithm accuracy and expand its application range, this paper tries to improve the SFA and studies the modified cubic surface fitting algorithm CTSFA (Corrected Three Surface Fitting Algorithm), which is suitable for solving the initial value of continuous displacement. Bilinear interpolation and adjacent interpolation are used to analyze the gray level at any integer‐pixel position in the displacement matrix and the weight coefficient is given. The distance‐weighted method is used to approximate the true initial displacement value of the continuum, and the algorithm suitable for digital image processing is extended to the continuum displacement solution. The cubic surface expression of the CTSFA programmatic application is solved by the least squares method, and the correlation coefficient of the power basis function is calculated. In the computer simulation of speckle test, the comparison between CTSFA and SFA on the calculation results of linear and nonlinear displacement fields shows that the calculated amount of CTSFA is basically the same as that of SFA, but the calculation accuracy is doubled. The study of analysing the Brazilian splitting test using CTSFA and SFA reveals that CTSFA is better than SFA in observing the development of cracks.