Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage

2011 ◽  
Vol 3 (6) ◽  
pp. 729-744
Author(s):  
Zhengzheng Hu ◽  
Mansoor A Haider
Keyword(s):  
Finite Element ◽  
Articular Cartilage ◽  
Analytical Solution ◽  
Conjugate Gradient ◽  
Numerical Solutions ◽  
Elastic Inclusion ◽  
Algebraic Multigrid ◽  
Inclusion Problem ◽  
Inclusion Problems ◽  
Cpu Time

AbstractIn studying biomechanical deformation in articular cartilage, the presence of cells (chondrocytes) necessitates the consideration of inhomogeneous elasticity problems in which cells are idealized as soft inclusions within a stiff extracellular matrix. An analytical solution of a soft inclusion problem is derived and used to evaluate iterative numerical solutions of the associated linear algebraic system based on discretization via the finite element method, and use of an iterative conjugate gradient method with algebraic multigrid preconditioning (AMG-PCG). Accuracy and efficiency of the AMG-PCG algorithm is compared to two other conjugate gradient algorithms with diagonal preconditioning (DS-PCG) or a modified incomplete LU decomposition (Euclid-PCG) based on comparison to the analytical solution. While all three algorithms are shown to be accurate, the AMG-PCG algorithm is demonstrated to provide significant savings in CPU time as the number of nodal unknowns is increased. In contrast to the other two algorithms, the AMG-PCG algorithm also exhibits little sensitivity of CPU time and number of iterations to variations in material properties that are known to significantly affect model variables. Results demonstrate the benefits of algebraic multigrid preconditioners for the iterative solution of assembled linear systems based on finite element modeling of soft elastic inclusion problems and may be particularly advantageous for large scale problems with many nodal unknowns.

Self-consistent estimates for the viscoelastic creep compliances of composite materials

1978 ◽  
Vol 359 (1697) ◽  
pp. 251-273 ◽  
Keyword(s):  
Composite Materials ◽  
Numerical Solutions ◽  
The Self ◽  
Inclusion Problem ◽  
Inclusion Problems ◽  
Viscoelastic Creep ◽  
Self Consistent ◽  
Title Problem ◽  
Consistent Method ◽  
Selection Of

In this paper the viscoelastic creep compliances of various composites are estimated by the self-consistent method. The phases may be arbitrarily anisotropic and in any concentrations but we demand that one of the phases be a matrix and the remaining phases consist of ellipsoidal inclusions. The theory is succinctly formulated with the help of Stieltjes convolutions. In order to solve the title problem, we first solve the misfitting viscoelastic inclusion problem. Numerical solutions are given for a selection of inclusion problems and for two common composite materials, namely an isotropic dispersion of spheres, and a uni-directional fibre reinforced material.

An Investigation of Dynamic Pulse Buckling of Thick Rings

1992 ◽  
Vol 59 (3) ◽  
pp. 615-621 ◽  
Author(s):  
N. G. Pegg
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Numerical Solutions ◽  
Nonlinear Finite Element ◽  
Finite Element Analyses ◽  
Elastoplastic Behavior ◽  
Impulse Load ◽  
Pulse Buckling ◽  
Comparison Of The Results ◽  
Dynamic Pulse

The occurrence of dynamic buckling of thick rings responding to an impulse load is investigated by analytical and numerical solutions to the equation of motion and by nonlinear finite element analyses. An extension to the linearized analytical solution is made using a finite difference scheme which incorporates a nonlinear moment-curvature relationship to model the effects of elastoplastic behavior and strain-rate reversal on the buckle formation. The finite element solution to the problem is formulated with the nonlinear code, ADINA. A comparison of the results shows that the numerical solutions (and, in particular, the ADINA solution) predict a significant reduction in the amplitude of buckling response and an increase in the predominant wavelength of response with time, in comparison to the linear analytical solution. A limited comparison to published experimental results of dynamic pulse buckling of thick rings is also given.

scholarly journals Spectral Analysis of Large Finite Element Problems by Optimization Methods

1994 ◽  
Vol 1 (6) ◽  
pp. 529-540 ◽  
Author(s):  
Luca Bergamaschi ◽  
Giuseppe Gambolati ◽  
Giorgio Pini
Keyword(s):  
Finite Element ◽  
Conjugate Gradient ◽  
Cost Effective ◽  
Optimization Methods ◽  
Optimal Choice ◽  
Lanczos Method ◽  
Asymptotic Convergence ◽  
Cpu Time ◽  
The One ◽  
Asymptotic Convergence Rate

Recently an efficient method for the solution of the partial symmetric eigenproblem (DACG, deflated-accelerated conjugate gradient) was developed, based on the conjugate gradient (CG) minimization of successive Rayleigh quotients over deflated subspaces of decreasing size. In this article four different choices of the coefficientβkrequired at each DACG iteration for the computation of the new search directionPkare discussed. The “optimal” choice is the one that yields the same asymptotic convergence rate as the CG scheme applied to the solution of linear systems. Numerical results point out that the optimalβkleads to a very cost effective algorithm in terms of CPU time in all the sample problems presented. Various preconditioners are also analyzed. It is found that DACG using the optimalβkand (LLT)−1as a preconditioner, L being the incomplete Cholesky factor of A, proves a very promising method for the partial eigensolution. It appears to be superior to the Lanczos method in the evaluation of the 40 leftmost eigenpairs of five finite element problems, and particularly for the largest problem, with size equal to 4560, for which the speed gain turns out to fall between 2.5 and 6.0, depending on the eigenpair level.

Application of the u-p Finite Element Method to the Study of Articular Cartilage

1991 ◽  
Vol 113 (4) ◽  
pp. 397-403 ◽  
Author(s):  
Jennifer S. Wayne ◽  
Savio L.-Y. Woo ◽  
Michael K. Kwan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Articular Cartilage ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Fluid Pressure ◽  
Numerical Procedure ◽  
Solid Matrix ◽  
Confined Compression ◽  
Element Method

The finite element method using the principle of virtual work was applied to the biphasic theory to establish a numerical routine for analyses of articular cartilage behavior. The matrix equations that resulted contained displacements of the solid matrix (u) and true fluid pressure (p) as the unknown variables at the element nodes. Both small and large strain conditions were considered. The algorithms and computer code for the analysis of two-dimensional plane strain, plane stress, and axially symmetric cases were developed. The u-p finite element numerical procedure demonstrated excellent agreement with available closed-form and numerical solutions for the configurations of confined compression and unconfined compression under small strains, and for confined compression under large strains. The model was also used to examine the behavior of a repaired articular surface. The differences in material properties between the repair tissue and normal cartilage resulted in significant deformation gradients across the repair interface as well as increased fluid efflux from the tissue.

Finite Element Resistivity Forward Modeling Using Algebraic Multigrid Preconditioned Conjugate Gradient Method

2012 ◽  
Vol 249-250 ◽  
pp. 792-797
Author(s):  
Gui Hong Zou ◽  
Hua Qing Liang
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Forward Modeling ◽  
Algebraic Multigrid ◽  
Preconditioned Conjugate Gradient ◽  
Uniform Mesh ◽  
Smoothed Aggregation

An algebraic multigrid by smoothed aggregation preconditioned conjugate gradient method is developed to solve the liner system arising from 3-D direct current finite element resistivity forward modeling. The algorithm combines the efficiency of algebraic multigrid method and the stability of conjugate gradient method. Algebraic multigrid by smoothed aggregation keep in high-efficiency while simulation using local quasi-uniform mesh and its convergence effect will reduce while numerical modeling using anisotropic stretched grids. However tensor product non-equidistant mesh, a kind of anisotropic stretched grids, is often used in 3-D direct current resistivity forward modeling. In order to improve this situation, a factor is added to guide correct aggregation. Consequently, a typical example is used to prove that the improvement is the right. Finally, it is natural to conclude that the algorithm suggested in this paper is efficient and robust whether simulation using local quasi-uniform mesh or tensor product non-equidistant mesh

Particular Solutions, Stress Singularities, and Stress Decay

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Asymptotic Solution ◽  
Numerical Solutions ◽  
Stress Singularities ◽  
Elastic Materials ◽  
Element Scheme ◽  
Regular Elements ◽  
Linear Elastic ◽  
Other Information

Not all boundary value problems are amenable to a simple analytical solution. This is particularly the case when the geometry of the boundary on which the boundary conditions are specified contains a corner. A crack tip is a special corner. When an analytical solution for the entire region is not available, asymptotic solutions near the corner can be obtained which provide useful information on the nature of stress singularities at the corner. They also provide more accurate numerical solutions by a finite element scheme in which the asymptotic solution at the corner is employed in a special element at the corner with regular elements elsewhere (see, for example, Stolarski and Chiang, 1989). Other information that can be obtained by an asymptotic analysis is the decay factor of stress at a large distance from a point at which a self-equilibrated load is applied (Crafter, et al., 1993). For singularities that arise in non-linear elastic materials the reader is referred to the book by Antman (1995). As in Chapter 8 most solutions can be expressed in a real form with the aid of identities presented in Chapters 6 and 7.

scholarly journals Theoretical analysis and experimental research on multi-layer elastic damping track structure

2021 ◽  
Vol 13 (2) ◽  
pp. 168781402199497
Author(s):  
Guanghui Xu ◽  
Shengkai Su ◽  
Anbin Wang ◽  
Ruolin Hu
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Finite Element Simulation ◽  
Reaction Force ◽  
Vibration Reduction ◽  
Vertical Direction ◽  
Propagation Path ◽  
Track Structure ◽  
Test Results ◽  
Element Simulation

The increase of axle load and train speed would cause intense wheelrail interactions, and lead to potential vibration related problems in train operation. For the low-frequency vibration reduction of a track system, a multi-layer track structure was proposed and analyzed theoretically and experimentally. Firstly, the analytical solution was derived theoretically, and followed by a parametric analysis to verify the vibration reduction performance. Then, a finite element simulation is carried out to highlight the influence of the tuned slab damper. Finally, the vibration and noise tests are performed to verify the results of the analytical solution and finite element simulation. As the finite element simulation indicates, after installation of the tuned slab damper, the peak reaction force of the foundation can be reduced by 60%, and the peak value of the vertical vibration acceleration would decrease by 50%. The vibration test results show that the insertion losses for the total vibration levels are 13.3 dB in the vertical direction and 21.7 dB in the transverse direction. The noise test results show that the data of each measurement point is smoother and smaller, and the noise in the generating position and propagation path can be reduced by 1.9 dB–5.5 dB.

scholarly journals Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

2020 ◽  
Vol 10 (1) ◽  
pp. 450-476
Author(s):  
Radu Ioan Boţ ◽  
Sorin-Mihai Grad ◽  
Dennis Meier ◽  
Mathias Staudigl
Keyword(s):  
Dynamical Systems ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Minimum Norm ◽  
Monotone Inclusion ◽  
Inclusion Problems ◽  
Monotone Inclusion Problem ◽  
Maximally Monotone Operators

Abstract In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

scholarly journals Biomechanical characteristics of tibio-femoral joint after partial medial meniscectomy in different flexion angles: a finite element analysis

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Xiaohui Zhang ◽  
Shuo Yuan ◽  
Jun Wang ◽  
Bagen Liao ◽  
De Liang
Keyword(s):  
Finite Element ◽  
Articular Cartilage ◽  
Finite Element Model ◽  
Medial Meniscus ◽  
Lateral Meniscus ◽  
Element Model ◽  
Magnetic Resonance Images ◽  
Biomechanical Analysis ◽  
Joint Stress ◽  
Tibiofemoral Joint

Abstract Background Recent studies have pointed out that arthroscopy, the commonly-used surgical procedure for meniscal tears, may lead to an elevated risk of knee osteoarthritis (KOA). The biomechanical factors of KOA can be clarified by the biomechanical analysis after arthroscopic partial meniscectomy (APM). This study aimed to elucidate the cartilage stress and meniscus displacement of the tibiofemoral joint under flexion and rotation loads after APM. Methods A detailed finite element model of the knee bone, cartilage, meniscus, and major ligaments was established by combining computed tomography and magnetic resonance images. Vertical load and front load were applied to simulate different knee buckling angles. At the same time, by simulating flexion of different degrees and internal and external rotations, the stresses on tibiofemoral articular cartilage and meniscus displacement were evaluated. Results Generally, the contact stress on both the femoral tibial articular cartilage and the meniscus increased with the increased flexion degree. Moreover, the maximum stress on the tibial plateau gradually moved backward. The maximum position shift value of the lateral meniscus was larger than that of the medial meniscus. Conclusion Our finite element model provides a realistic three-dimensional model to evaluate the influence of different joint range of motion and rotating tibiofemoral joint stress distribution. The decreased displacement of the medial meniscus may explain the higher pressure on the knee components. These characteristics of the medial tibiofemoral joint indicate the potential biomechanical risk of knee degeneration.

Numerical solutions for strain-softening surrounding rock under three-dimensional principal stress condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sheng Yu-ming ◽  
Li Chao ◽  
Xia Ming-yao ◽  
Zou Jin-feng
Keyword(s):  
Finite Element ◽  
Principal Stress ◽  
Stress Condition ◽  
Strain Softening ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Strength Criterion ◽  
Surrounding Rock ◽  
Flow Rule ◽  
Finite Element Software

Abstract In this study, elastoplastic model for the surrounding rock of axisymmetric circular tunnel is investigated under three-dimensional (3D) principal stress states. Novel numerical solutions for strain-softening surrounding rock were first proposed based on the modified 3D Hoek–Brown criterion and the associated flow rule. Under a 3D axisymmetric coordinate system, the distributions for stresses and displacement can be effectively determined on the basis of the redeveloped stress increment approach. The modified 3D Hoek–Brown strength criterion is also embedded into finite element software to characterize the yielding state of surrounding rock based on the modified yield surface and stress renewal algorithm. The Euler implicit constitutive integral algorithm and the consistent tangent stiffness matrix are reconstructed in terms of the 3D Hoek–Brown strength criterion. Therefore, the numerical solutions and finite element method (FEM) models for the deep buried tunnel under 3D principal stress condition are presented, so that the stability analysis of surrounding rock can be conducted in a direct and convenient way. The reliability of the proposed solutions was verified by comparison of the principal stresses obtained by the developed numerical approach and FEM model. From a practical point of view, the proposed approach can also be applied for the determination of ground response curve of the tunnel, which shows a satisfying accuracy compared with the measuring data.

Sign in / Sign up

Export Citation Format

Share Document