Linear Elastic Problems Research Articles

Elasticity imaging using physics-informed neural networks: Spatial discovery of elastic modulus and Poisson's ratio.

Elasticity imaging is a technique that discovers the spatial distribution of mechanical properties of tissue using deformation and force measurements under various loading conditions. Given the complexity of this discovery, most existing methods approximate only one material parameter while assuming homogeneous distributions for the others. We employ physics-informed neural networks (PINN) in linear elasticity problems to discover the space-dependent distribution of both elastic modulus (E) and Poisson's ratio (ν) simultaneously, using strain data, normal stress boundary conditions, and the governing physics. We validated our model on three examples. First, we experimentally loaded hydrogel samples with embedded stiff inclusions, representing tumorous tissue, and compared the approximations against ground truth determined through tensile tests. Next, using data from finite element simulation of a rectangular domain containing a stiff circular inclusion, the PINN model accurately localized the inclusion and estimated both E and ν. We observed that in a heterogeneous domain, assuming a homogeneous ν distribution increases estimation error for stiffness as well as the area of the stiff inclusion, which could have clinical importance when determining size and stiffness of tumorous tissue. Finally, our model accurately captured spatial distribution of mechanical properties and the tissue interfaces on data from another computational model, simulating uniaxial loading of a rectangular hydrogel sample containing a human brain slice with distinct gray matter and white matter regions and complex geometrical features. This elasticity imaging implementation has the potential to be used in clinical imaging scenarios to reliably discover the spatial distribution of mechanical parameters and identify material interfaces such as tumors. STATEMENT OF SIGNIFICANCE: Our work is the first implementation of physics-informed neural networks to reconstruct both material parameters - Young's modulus and Poisson's ratio - and stress distributions for isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using experimental measurements and synthetic data generated using finite element modeling. Our method can be implemented in clinical elasticity imaging scenarios to improve diagnosis of tumors and for mechanical characterization of biomaterials and biological tissues in a minimally invasive manner.

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Computer Aided Design (CAD) is widely used in the creation and optimization of various industrial systems and processes. Transforming a CAD geometry into a computational discretization that be used to solve PDEs requires care and a deep knowledge of the selected computational method. In this article, we present a novel integrated collocation scheme based on smart clouds. It allows us to transform a CAD geometry into a complete point collocation model, aware of the base geometry, with minimum effort. For this process, only the geometry of the domain, in the form of a STEP file, and the boundary conditions are needed. We also introduce an adaptive refinement process for the resultant smart cloud using an a posteriori error indication. The scheme can be applied to any 2D or 3D geometry, to any PDE and can be applied to most point collocation approaches. We illustrate this with the meshfree Generalized Finite Difference (GFD) method applied to steady linear elasticity problems. We further show that each step of this process, from the initial discretization to the refinement strategy, is connected and is affected by the approach selected in the previous step, thus requiring an integrated scheme where the whole solution process should be considered at once. • CAD files are used to generate smart point clouds without any intermediate step. • Exact normal vectors and boundary conditions are based on the CAD geometry. • ZZ-type error indicators are the basis for model adaptivity and are quick to compute. • Model adaptivity speeds up the convergence rate of the solution. • Model adaptivity is based on the exact CAD geometry.

A hybridized weak Galerkin finite element scheme for linear elasticity problem

In this paper, we present a hybridized weak Galerkin finite element method for a class of a linear elasticity model. The hybridized weak Galerkin finite element method is an improvement of the standard weak Galerkin method by introducing Lagrange multipliers, which relaxes certain constraints and inherits its flexibility. By applying Schur complement technique, the degrees of freedom is reduced and the calculation efficiency is improved. The optimal convergence orders of error estimates are achieved. Some numerical experiments are presented to illustrate the theoretical analysis.

Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios

Recently, a class of machine learning methods called physics-informed neural networks (PINNs) has been proposed and gained prevalence in solving various scientific computing problems. This approach enables the solution of partial differential equations (PDEs) via embedding physical laws into the loss function. Many inverse problems can be tackled by simply combining the data from real life scenarios with existing PINN algorithms. In this paper, we present a multi-task learning method using uncertainty weighting to improve the training efficiency and accuracy of PINNs for inverse problems in linear elasticity and hyperelasticity. Furthermore, we demonstrate an application of PINNs to a practical inverse problem in structural analysis: prediction of external loads of diverse engineering structures based on limited displacement monitoring points. To this end, we first determine a simplified loading scenario at the offline stage. By setting unknown boundary conditions as learnable parameters, PINNs can predict the external loads with the support of measured data. When it comes to the online stage in real engineering projects, transfer learning is employed to fine-tune the pre-trained model from offline stage. Our results show that, even with noisy gappy data, satisfactory results can still be obtained from the PINN model due to the dual regularization of physics laws and prior knowledge, which exhibits better robustness compared to traditional analysis methods. Our approach is capable of bridging the gap between various structures with geometric scaling and under different loading scenarios, and the convergence of training is also greatly accelerated through not only the layer freezing but also the multi-task weight inheritance from pre-trained models, thus making it possible to be applied as surrogate models in actual engineering projects.

Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains

A fuzzy arithmetic framework for the efficient possibilistic propagation of shape uncertainties based on a novel fuzzy edge detection method is introduced. The shape uncertainties stem from a blurred image that encodes the distribution of two phases in a composite material. The proposed framework employs computational homogenisation to upscale the shape uncertainty to a effective material with fuzzy material properties. For this, many samples of a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying mapping from shape parametrisation to the upscaled material behaviour, a diffeomorphism is constructed by generating an appropriate family of meshes via transformation of a reference mesh. The shape uncertainty is then propagated to measure the distance of the upscaled material to the isotropic and orthotropic material class. Finally, the fuzzy effective material is used to compute bounds for the average displacement of a non-homogenized material with uncertain star-shaped inclusion shapes.

A coupling approach for linear elasticity problems with spatially non-coincident discretized interfaces

We present a new method for coupled linear elasticity problems whose finite element discretization may lead to spatially non-coincident discretized interfaces. Our approach combines the classical Dirichlet–Neumann coupling formulation with a new set of discretized interface conditions obtained through Taylor series expansions. We show that these conditions ensure linear consistency of the coupled finite element solution. We then formulate an iterative solution method for the coupled discrete system and apply the new coupling approach to two representative settings for which we also provide several numerical illustrations. The first setting is a mesh-tying problem in which both coupled structures have the same Lamé parameters whereas the second setting is an interface problem for which the Lamé parameters in the two coupled structures are different.

Boundary element procedure for computation of internal directional derivatives in homogeneous Laplace's problems solved by the finite element method

Using the standard approach of the Finite Element Method, numerical values of the spatial derivatives of the primal variable present low accuracy. It is due to the order reduction in the interpolation functions used for the approximation of the derivatives. Aiming to overcome this problem, this work uses the integral equation of the Boundary Element Method to recalculate new values of internal variables, using the nodal boundary values obtained by the Finite Element solution. The mathematical foundation for this procedure is based on the idea that the boundary integral equation is equivalent to a weighted residual sentence, and its reuse implies a new minimization of numerical errors. This strategy was previously used within the Boundary Element context to successfully recalculate nodal variables in scalar problems governed by Laplace and Poisson Equations, also used successfully for solving linear elastic problems expressed by Navier's Equation. Here, to confirm the consistency of the proposed model, computational tests are performed, in which the standard FEM results are compared with those obtained by the proposed procedure. Internal potential derivatives and also internal potential values were recalculated. To assess the quality of the solution, the results are compared to a benchmark.

The elastic deformation process of bimetallic objects

In industrial problems linear elasticity simulation has been known to be successfully solved by the Finite Element Methods (FEMs). However, a mathematical model as well as a numerical solution program that can be applied to the deformation simulation of bimetallic materials are still the challenges in this research area. In this paper, we investigate a mathematical model for the linear elastic deformation of bimetallic structures. By reducing the linear elasticity problem to its weak form, we apply the continuous Galerkin approximation to obtain the discrete problem. The existence and the uniqueness of the solution of the weak problem are clearly demonstrated, ensuring the well-posedness of the problem. In addition, the numerical simulations with bimetallic materials have been built on Freefem[Formula: see text] software that illustrate the efficiency and reliability of the numerical method.

An enhanced extended isogeometric analysis with strong imposition of essential boundary conditions for crack problems using B++ splines

We present the extended isogeometric analysis (XIGA) based on B++ splines for linear elastic fracture problems. The proposed approach converts the control points associated with crack elements into collocation points on the crack surface. The opening of the crack surface is characterized by double-layer collocation points whereas the tip singularity is simulated by additional collocation points. In doing so, the additional control points associated with tip elements are replaced by the additional collocation points on the crack surface, which reduces the number of unknowns. The collocation points on the entire crack surface satisfy the Kronecker Delta property, which allows directly imposing Dirichlet boundary condition on the crack surface. A posteriori error estimation technique is also presented in this paper. The values of the normalised stress intensity factors (SIFs) are calculated by interaction integral technique. Finally, the suggested approach is verified by 2D benchmark problems.

H-Adaptive radial basis function finite difference method for linear elasticity problems

In this research work, the radial basis function finite difference method (RBF-FD) is further developed to solve one- and two-dimensional boundary value problems in linear elasticity. The related differentiation weights are generated by using the extended version of the RBF utilizing a polynomial basis. The type of the RBF is restricted to polyharmonic splines (PHS), i.e., a combination of the odd m-order PHS phi (r)=r^m with additional polynomials up to degree p will serve as the basis. Furthermore, a new residual-based adaptive point-cloud refinement algorithm will be presented and its numerical performance will be demonstrated. The computational efficiency of the PHS RBF-FD approach is tested by means of the relative errors measured in ell _2-norm on several representative benchmark problems with smooth and non-smooth solutions, using h-adaptive, uniform, and quasi-uniform point-cloud refinement.

On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

We investigate the possibility to determine the divergence-free displacement uindependently from the pressure reaction p for a class of boundary-value problems in incompressible linear elasticity. If not possible, we investigate if it is possible to determine it pressure robustly, i.e. pollution free from the pressure reaction.For convex domains there is but one variational boundary value problem among the investigated that allows the independent determination. It is the one with essential no-penetration conditions combined with homogeneous tangential traction conditions.Further, for the investigated cases, the weakly divergence-free displacement can be computed pressure robustly provided the total body force is decomposed into its direct sum of divergence- and rotation-free components using the Helmholtz decomposition. The elasticity problem is solved using these components as separate right-hand sides. The total solution is obtained using the superposition principle.We employ a (u,p) higher-order finite element formulation with discontinuous pressure elements. It is inf–sup stable for polynomial degree p≥2 but not pressure robust by itself. We propose a three step procedure to solve the elasticity problem which includes by the Helmholtz decomposition of the total body force. The extra cost for the three-step procedure is essentially the cost for the Helmholtz decomposition of the assembled total body force, and the small cost of solving the elasticity problem with one extra right-hand side. The results are corroborated by theoretical derivations as well as numerical results.

Phase-field fracture modeling for large structures

The phase-field modeling framework for linear elastic fracture mechanics problems is modified for the purpose of analyzing crack growth in large structures. The phase-field approach to fracture replaces sharp crack surfaces with a diffuse fracture zone that represent the traction-free crack faces. The diffuse fracture zone is characterized by a length scale that appears prominently in the governing partial differential equation that governs the evolution of the phase-field variable. Within numerical calculations the diffuse fracture zone must be resolved with a sufficient number of degrees of freedom in order to obtain accurate solutions. Additionally, all material fracture problems possess a physical process zone length scale that scales with the ratio of the fracture energy to the square of the material strength. For the vast majority of problems governed by linear elastic fracture mechanics, this physical process zone length scale is small, for example it is on the order of microns for aluminum alloys. If the material strength is to be captured, then the most commonly used formulation of the phase-field approach to fracture ties the diffuse phase-field crack length scale to the physical process zone length scale. This presents a challenge for extending such models to large structures that may be on the order of meters in size due to the prohibitive meshing requirements for the phase-field crack length scale. This is especially the case for three-dimensional problems. To address this problem a new form of the phase-field degradation function is introduced that allows for a decoupling of the phase-field length scale from the physical process zone length scale. The behavior and limitations of this new formulation are discussed and illustrated with a series of numerical test cases.

The effect of hydrology and crevasse wall contact on calving

Abstract. Calving is one of the main controls on the dynamics of marine ice sheets. We solve a quasi-static linear elastic fracture dynamics problem, forced by a viscous pre-stress describing the stress state in the ice prior to the introduction of a crack, to determine conditions under which an ice shelf can calve for a variety of different surface hydrologies. Extending previous work, we develop a boundary-element-based method for solving the problem, which enables us to ensure that the faces of crevasses are not spuriously allowed to penetrate into each other in the model. We find that a fixed water table below the ice surface can lead to two distinct styles of calving, one of which involves the abrupt unstable growth of a crack across a finite thickness of unbroken ice that is potentially history-dependent, while the other involves the continuous growth of the crack until the full ice thickness has been penetrated, which occurs at a critical combination of extensional stress, water level and ice thickness. We give a relatively simple analytical calving law for the latter case. For a fixed water volume injected into a surface crack, we find that complete crack propagation almost invariably happens at realistic extensional stresses if the initial crack length exceeds a shallow threshold, but we also argue that this process is more likely to correspond to the formation of a localized, moulin-like slot that permits drainage, rather than a calving event. We also revisit the formation of basal cracks and find that, in the model, they invariably propagate across the full ice shelf at stresses that are readily generated near an ice shelf front. This indicates that a more sophisticated coupling of the present model (which has been used in a very similar form by several previous authors) needs modification to incorporate the effect of torques generated by buoyantly modulated shelf flexure in the far field.

A modified weak Galerkin finite element method for the linear elasticity problem in mixed form

In this paper, we present and analyze the modified weak Galerkin (MWG) finite element method to the linear elasticity problem in a mixed form. Comparing with weak Galerkin (WG) finite element method, the structure of modified weak Galerkin (MWG) finite element method is especial. The main idea of MWG finite element method for elasticity problem is to express to boundary stress tensor defined on the partition boundary in terms of the average of the interior stress tensor defined inside partition elements. This can substantially reduce the degrees of freedom of the linear system. The well-posedness of the solution and inf–sup condition are investigated in this paper. We obtain the optimal error estimates in both energy norm and L2 norm. Finally, some numerical experiments are exhibited to illustrate the efficiency and accuracy of the modified weak Galerkin finite element method.

A physics‐based model reduction approach for node‐to‐segment contact problems in linear elasticity

AbstractThe article presents a novel model order reduction method for mechanical problems in linear elasticity with nonlinear contact conditions. Recently, we have proposed an efficient reduction scheme for the node‐to‐node formulation [Manvelyan et al., Comput Mech 68, 1283–1295 (2021)] that leads to linear complementarity problems (LCPs). Here, we enhance the underlying contact problem to a node‐to‐segment formulation, which leads to quadratic inequalities as constraints. The adjoint system corresponds to a nonlinear complementarity problem that describes the Lagrange multiplier. The latter is solved by a Newton‐type iteration based on a LCP solver in each time step. Since the maximal set of potential contact nodes is predefined, an additional substructuring by Craig–Bampton can be performed. This contact treatment turns out to be necessary and allows exclusion of the Lagrange multipliers and the nodal displacements at contact from reduction. The numerical solutions of the reduced contact problem achieve high accuracy and the dynamic contact carries over the behavior of the full order model. Moreover, if the contact area is small compared to the overall structure, the reduction scheme performs very efficiently. The performance of the resulting reduction method is assessed on two 2D computational examples from linear elasticity.

Stable generalized finite element method (SGFEM) for three-dimensional crack problems

This paper proposes a stable generalized finite element method (SGFEM) for the linear 3D elasticity problem with cracked domains. Conventional material-independent branch functions serve as singular enrichments. We prove that the proposed SGFEM with the geometric enrichment scheme yields the optimal order of convergence in the energy norm, O(h), for fully 3D elasticity planar crack problems; h is the mesh parameter. To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be O(h^{-2}) (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.

Parallel Scalability of Three-Level FROSch Preconditioners to 220000 Cores using the Theta Supercomputer

The parallel performance of the three-level fast and robust overlapping Schwarz (FROSch) preconditioners is investigated for linear elasticity. The FROSch framework is part of the Trilinos software library and contains a parallel implementation of different preconditioners with energy minimizing coarse spaces of generalized Dryja--Smith--Widlund type. The three-level extension is constructed by a recursive application of the FROSch preconditioner to the coarse problem. In this paper, the additional steps in the implementation in order to apply the FROSch preconditioner recursively are described in detail. Furthermore, it is shown that no explicit geometric information is needed in the recursive application of the preconditioner. In particular, the rigid body modes, including the rotations, can be interpolated on the coarse level without additional geometric information. Parallel results for a three-dimensional linear elasticity problem obtained on the Theta supercomputer (Argonne Leadership Computing Facility, Argonne, IL) using up to 220 000 cores are discussed and compared to results obtained on the SuperMUC-NG supercomputer (Leibniz Supercomputing Centre, Garching, Germany). Notably, it can be observed that a hierarchical communication operation in FROSch related to the coarse operator starts to dominate the computing time on Theta, which has a dragonfly interconnect, for 100 000 message passing interface (MPI) ranks or more. The same operation, however, scales well and stays within the order of a second in all experiments performed on SuperMUC-NG, which uses a fat tree network. Using hybrid MPI/OpenMP parallelization, the onset of the MPI communication problem on Theta can be delayed. Further analysis of the performance of FROSch on large supercomputers with dragonfly interconnects will be necessary.

Primal hybrid finite element method for the linear elasticity problem

In this manuscript, we study a primal hybrid finite element method for two dimensional linear elasticity problem. We derive a priori error estimates for both primal and hybrid variables. The rate of convergence of the method is independent of the Lamé parameters, which illustrates the robustness of the method. Numerical experiments are presented to validate the theoretical findings.

Pointwise A Posteriori Error Control of Discontinuous Galerkin Methods for Unilateral Contact Problems

Abstract In this article, we derive a reliable and efficient a posteriori error estimator in the supremum norm for a class of discontinuous Galerkin (DG) methods for the frictionless unilateral contact problem between two elastic bodies. The proposed error estimator generalizes the basic residual type estimators for the linear problems in linear elasticity taking into account the nonlinearity on a part of the boundary. The analysis hinges on the super- and sub-solutions constructed by modifying the discrete solution appropriately, and it is carried out in a unified manner which holds for several DG methods. The terms arising from the contact stresses in the error estimator vanish on the discrete full contact set. We illustrate the performance of the proposed error estimator via several numerical experiments in two dimensions.

On tracking arbitrary crack path with complex variable meshless methods

This study presents a numerical modelling framework based on complex variable meshless methods, which can accurately and efficiently track arbitrary crack paths in two-dimensional linear elastic solids. The key novelty of this work is that the proposed meshless modelling scheme enables a direct element-free approximation for the solutions of linear elastic fracture mechanics problems. The complex variable moving least-squares approximation with a group of simple complex polynomial basis is applied to implement this meshless model, with which the fracture problems with both stationary or progressive cracks are considered and studied. The effects of choosing different definitions of weighted complex variable error norm and different forms of complex polynomial basis on the computational accuracy of crack tip fields and crack paths prediction are analysed and discussed. Five benchmark numerical examples were studied to demonstrate the superiority of the present complex variable meshless framework over a standard element-free Galerkin method in tracking arbitrary crack paths in two-dimensional elastic solids.