Elastic Problems Research Articles

Isotropic elasticity of dilatational conical inclusion. An analytical approach

In this article, the analytical solution for the isotropic elasticity problem for the cone-shaped inclusion with dilatational eigenstrain in an infinite medium is given. The conical inclusion is modeled by dilatational infinitesimally thin circular disks distributed continuously along the cone z-axis with the radii of the disks being proportional to z-coordinate. The displacements, strains, and stresses of the conical inclusion are given in spherical coordinates with the origin in the cone-apex in the form of the series with Legendre polynomials. The maps of the displacements and the stresses are presented. A comparison of the displacements of the conical inclusion with the displacements of the finite cylindrical and hemispherical inclusions is provided. It is also shown, as expected, that the energy of the dilatational inclusion does not depend on its shape. In Discussion section, the specific features of an “hourglass” dipole inclusion consisting of two conical inclusions with different sign eigenstrains are demonstrated.

Ocean bottom dual-sensorQ-compensated elastic least-squares reverse time migration based on acoustic and separated-viscoelastic coupling equations

The ocean bottom dual-sensor (OBD) observation system places water detectors (pressure sensors) and land detectors on the irregular seabed interface to receive acoustic pressure fields and elastic wavefields, respectively. These data are usually applied to effectively remove ghost waves and ringing interference in the seawater layer, rather than being jointly used for migration. We propose an ocean bottom dual-sensor viscoacoustic and separated-viscoelastic coupled Q-compensated least-squares reverse time migration (OBD-VSV- QLSRTM) in curvilinear coordinates. In this method, to solve the Q-compensated elastic inverse problem, we jointly use the recorded acoustic pressure and elastic multicomponent seismic data from the OBD observation system. A new misfit function in the sense of least-squares inversion is constructed based on viscoacoustic and separated viscoelastic wavefields. Acoustic and separated viscoelastic equations are used to describe the acoustic wavefield in the acoustic medium and the Q-compensated P and S wavefields in the underlying viscoelastic medium, respectively. On the acoustic-viscoelastic interface, we implement a stable governing equation to ensure continuous and steady transmission of Q-compensated stresses in the subseabed and pressure in the seawater. To address the irregular seabed problem, we mesh the acoustic-viscoelastic models onto curvilinear grids and apply coordinate transformations to map the models and equations to a new curvilinear coordinate system. We derive viscoacoustic and separated-viscoelastic coupled adjoint and demigration operators in the curvilinear coordinates, enabling linear acoustic-viscoelastic modeling and stable receiver-side back-propagated wavefield continuation. Numerical examples conducted on two typical acoustic-viscoelastic models demonstrate that our proposed OBD-VSV- QLSRTM in curvilinear coordinates can produce highly accurate P- and S-wave images efficiently.

Reconstructing relaxed configurations in elastic bodies: Mathematical formulations and numerical methods for cardiac modeling

Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield’s seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier’s iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja–Smith–Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios where Sellier’s algorithm proves inadequate.

Solution of planar elastic stress problems using stress basis functions

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ p , is any stress in equilibrium with the loading. The second part, σ h is a self-equilibrated stress field on the unloaded body. In both methods, σ h is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.

A Robust and High Precision Algorithm for Elastic Scattering Problems from Cornered Domains

A Robust and High Precision Algorithm for Elastic Scattering Problems from Cornered Domains

M‐PINN: A mesh‐based physics‐informed neural network for linear elastic problems in solid mechanics

AbstractPhysics‐informed neural networks (PINNs) have emerged as a promising approach for solving a wide range of numerical problems. Nevertheless, conventional PINNs frequently face challenges in model convergence and stability when optimizing complex loss functions containing complex gradients. In this study, a new mesh‐based PINN method, called M‐PINN, is proposed drawing the ideas of the finite element method (FEM). By partitioning the solution domain into several subdomains and incorporating finite element data distribution constraints to the prior estimates of the predicted data distribution of PINN on the solution domain, the M‐PINN approach effectively reduces the optimization difficulty of conventional PINNs. Moreover, it is sometimes difficult to directly obtain precise boundary conditions in some practical applications. This method can be used to solve PINN problems with unknown boundary conditions, thus having wider applicability. In this study, the efficiency of M‐PINN was demonstrated through a standard 2D linear elastic solid mechanics simulation experiment, and its applicability was investigated in depth. The results indicate that the M‐PINN method outperforms traditional PINN and exhibits superior applicability and convergence, especially in cases involving unknown boundary conditions.

MechAgents: Large language model multi-agent collaborations can solve mechanics problems, generate new data, and integrate knowledge

Solving mechanics problems using numerical methods requires comprehensive intelligent capability of retrieving relevant knowledge and theory, constructing and executing codes, analyzing the results, a task that has thus far mainly been reserved for humans. While emerging AI methods can provide effective approaches to solve end-to-end problems, for instance via the use of deep surrogate models or various data analytics strategies, they often lack physical intuition since knowledge is baked into the parametric complement through training, offering less flexibility when it comes to incorporating mathematical or physical insights. By leveraging diverse capabilities of multiple dynamically interacting large language models (LLMs), we can overcome the limitations of conventional approaches and develop a new class of physics-inspired generative machine learning platform, here referred to as MechAgents. A set of AI agents can solve mechanics tasks, here demonstrated for elasticity problems, via autonomous collaborations. A two-agent team can effectively write, execute and self-correct code, in order to apply finite element methods to solve classical elasticity problems in various flavors (different boundary conditions, domain geometries, meshes, small/finite deformation and linear/hyper-elastic constitutive laws, and others). For more complex tasks, we construct a larger group of agents with enhanced division of labor among planning, formulating, coding, executing and criticizing the process and results. The agents mutually correct each other to improve the overall team-work performance in understanding, formulating and validating the solution. Our framework shows the potential of synergizing the intelligence of language models, the reliability of physics-based modeling, and the dynamic collaborations among diverse agents, opening novel avenues for automation of solving engineering problems.

A FETI B-differentiable equation method for elastic frictional contact problem with nonconforming mesh

A FETI B-differentiable equation method for elastic frictional contact problem with nonconforming mesh

A family of three-dimensional virtual elements for Hellinger-Reissner elasticity problems

We present a family of Virtual Element Methods for three-dimensional linear elasticity problems based on the Hellinger-Reissner variational principle. A convergence and stability analysis is developed. Moreover, using the hybridization technique and exploiting the information derived from this procedure, we show how to compute a better approximation for the displacement field. The numerical experiments confirm the theoretical predictions.

Stationary response of the system “Cylindrical shell – viscoelastic filler” to the effect of a moving load

The paper considers the stationary response of the system “cylindrical shell - viscoelastic filler” to the action of a moving load. The filler and the shell were assumed to be viscoelastic. Using the principle of analogy of viscoelastic and elastic problems (elastic-viscoelastic analogy), a stationary solution of the action of mobile loads on a shell with a viscoelastic filler is obtained. It is shown that this principle makes it possible to generalize the class of problems obtained for the case of viscoelastic media. Since the viscoelastic filler has damping properties, the inversion integrals have no singularities on the real axis, and the elements of the determinants in integrals become complex since in this problem in the image space, the Lame coefficients are complex and depend on the speed of the load movement, as well as the Fourier transform parameter. Numerical results are obtained, and an analysis is made.

Galerkin finite block method in solid mechanics

In this paper, the Galerkin Finite Block Method (GFBM) is proposed for the first time to solve two-dimensional elasticity problems with functionally graded materials (FGMs). The physical variables are approximated with double layer Chebyshev polynomials with unknown coefficients for two-dimensional elasticity. The main idea is to establish a set of linear algebraic equations by the Galerkin method using Chebyshev polynomials in the weak form of the partial differential equations and the boundary conditions. By introducing the mapping technique, a block with irregular shape of the boundary is transformed from the Cartesian coordinates (x, y) to the normalized coordinates (ξ, η), |ξ|⩽1, |η|⩽1. The continuity conditions of the displacement and traction on the interface between two blocks are introduced both in the weak form and strong form respectively. Several numerical examples are presented and comparisons are made with the Finite Element Method (FEM) including a cracked FGM sheet, in order to demonstrate the accuracy and efficiency of the GFBM.

A parameter-free mixed formulation for the Stokes equations and linear elasticity with strongly symmetric stress

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H(div;Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories.

Enhancing elasticity models with deep learning: A novel corrective source term approach for accurate predictions

With the recent wave of digitalization, specifically in the context of safety–critical applications, there has been a growing need for computationally efficient, accurate, generalizable, and trustworthy models. Physics-based models have traditionally been used extensively for simulating and understanding complex phenomena. However, these models though trustworthy and generalizable to a wide array of problems, are not ideal for real-time. To address this issue, the physics-based models are simplified. Unfortunately, these simplifications, like reducing the dimension of the problem (3D to 2D) or linearizing the highly non-linear characteristics of the problem, can degrade model accuracy. Data-driven models, on the other hand, can exhibit better computational efficiency and accuracy. However, they fail to generalize and operate as blackbox, limiting their acceptability in safety–critical applications. In the current article, we demonstrate how we can use a data-driven approach to correct for the two kinds of simplifications in a physics-based model. To demonstrate the methodology’s effectiveness, we apply the method to model several elasticity problems. The results show that the hybrid approach, which we call the corrective source term approach, can make erroneous physics-based models more accurate and certain. The hybrid model also exhibits superior performance in terms of accuracy, model uncertainty, and generalizability when compared to its end-to-end data-driven modeling counterpart.

On the Importance of the Recovery Procedure in the Semi-Analytical Solution for the Static Analysis of Curved Laminated Panels: Comparison with 3D Finite Elements.

The manuscript presents an efficient semi-analytical solution with three-dimensional capabilities for the evaluation of the static response of laminated curved structures subjected to general external loads. A two-dimensional model is presented based on the Equivalent Single Layer (ESL) approach, where the displacement field components are described with a generalized formulation based on a higher-order expansion along the thickness direction. The fundamental equations are derived from the Hamiltonian principle, and the solution is found by means of Navier's approach. Then, an efficient recovery procedure, derived from the three-dimensional elasticity equations and based on the Generalized Differential Quadrature (GDQ) method, is adopted for the derivation of the three-dimensional solution. Some examples of investigation are presented, where the numerical predictions of refined three-dimensional Finite-Element-based models are matched with a high level of accuracy. The model is validated for both straight and curved panels, taking into account different lamination schemes and load shapes. Furthermore, it is shown that the numerical solution to the elasticity problem in the recovery procedure is determining and accurately predicting the three-dimensional static response of the doubly-curved shell solid.

A Comparative Study on Two Mixed Least Squares Meshless Models with Improved SPH, MPS and CPM Methods to Solve Elasticity Problems

A Comparative Study on Two Mixed Least Squares Meshless Models with Improved SPH, MPS and CPM Methods to Solve Elasticity Problems

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities

Analysis of axisymmetric hollow cylinder under surface loading using variational principle

In this work, a variational principle–based approach has been adopted to analyze one of the classical linear elasticity problem of the axisymmetric cylinder under surface loading. The use of variational principle results in a set of governing partial differential equations with associated boundary conditions. The equations have been solved using the separation of variable approach and the Frobenius method. A general solution has been derived and used to solve two test cases. The proposed solution is capable of meeting all the boundary conditions. The solution has been validated by comparing it with a finite element–based numerical solution and considering a special limiting condition of a solid cylinder, for which results are available in the literature. Further various studies have been carried out to understand the robustness and limitation of the presented solution.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

Quantitative stability estimate for the inverse coefficients problem in linear elasticity

In this article we consider the inverse problem of reconstructing piece-wise Lamé coefficients from boundary measurements. We reformulate the inverse problem into a minimization one using a Kohn-Vogelius type functional. We study the stability of the parameters when the jump of the discontinuity is perturbed. Using tools of shape calculus, we give a quantitative stability result for local optimal solution. Dans cet article, nous considérons le problème inverse de reconstruction des coefficients de Lamé constants par morceaux à partir de mesures au bord. Nous reformulons le problème inverse en un problème de minimisation utilisant une fonctionnelle de type Kohn-Vogelius. Nous étudions la stabilité des paramètres lorsque le saut de la discontinuité est perturbé. En utilisant les outils du calcul de forme, nous donnons un résultat de stabilité quantitative pour une la solution optimale locale.

Maximum elastic scheduling of virtual machines in general graph cloud data center networks

ABSTRACT In this research, we pioneer a novel method to evaluate the maximum admissible load (MAL) for virtual machines (VMs) in physical machines (PMs) in data centre networks (DCNs), without restricting DCN topologies. This unique approach simplifies the issue into a single-source, multiple-sink maximum flow problem. It also resolves the maximum elastic scheduling problem by determining the optimal load for consistent growth without reassigning tasks. An effective strategy for these challenges is introduced and validated through extensive simulations.