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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E2: Control Theory and Mechanics".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 34388

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Dr. Isabelle Ramiere

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Guest Editor

CEA, DES, IRESNE, DEC, SESC, F-13108 Saint-Paul-lez-Durance, France
Interests: applied mathematics; numerical methods; computational physics; computational mechanics

Prof. Dr. Frédéric C. Lebon

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Guest Editor

CNRS, Laboratoire de Mécanique et d'Acoustique, Université Aix-Marseille, 13007 Marseille, France
Interests: structure mechanics; solid mechanics; computational mechanics; contact mechanics; modeling
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Dear Colleagues,

Efficient numerical solving of nonlinear solid mechanics problems is still a challenging issue which concerns various fields: nonlinear behavior, micromechanics, contact mechanics, damage, cracks propagation, rupture, etc. Numerical methods dedicated to such topics have been developed for many decades, but many fundamental and important challenges remain. In particular, multiscale methods which bridge different scales in time and space, efficient reduced-order models for variational inequalities or fully scalable nonlinear solvers, to name but a few.

For this Special Issue, we seek contributions which introduce or adapt advanced numerical methods for computational mechanics. Topics of interest include, but are not limited to, the following: adaptive mesh refinement, domain decomposition method, multiscale approaches for heterogeneous materials, reduced order modeling, efficient nonlinear solvers, parallel computing, contact mechanics, and crack initiation and/or propagation.

Dr. Isabelle Ramiere
Prof. Dr. Frédéric C. Lebon
Guest Editors

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Keywords

Advanced numerical method
Computational solid mechanics
Multiscale and adaptive approaches
Reduced order modeling
Efficient nonlinear solvers
Contact mechanics
Crack propagation

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Published Papers (11 papers)

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Editorial

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3 pages, 190 KiB

Open AccessEditorial

Advanced Numerical Methods in Computational Solid Mechanics

by Frédéric Lebon and Isabelle Ramière

Mathematics 2023, 11(6), 1512; https://doi.org/10.3390/math11061512 - 20 Mar 2023

Cited by 1 | Viewed by 2161

Abstract

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

Research

Jump to: Editorial

31 pages, 8003 KiB

Open AccessArticle

Proximity Effects in Matrix-Inclusion Composites: Elastic Effective Behavior, Phase Moments, and Full-Field Computational Analysis

by Louis Belgrand, Isabelle Ramière, Rodrigue Largenton and Frédéric Lebon

Mathematics 2022, 10(23), 4437; https://doi.org/10.3390/math10234437 - 24 Nov 2022

Cited by 3 | Viewed by 1639

Abstract

This work focuses on the effects of inclusion proximity on the elastic behavior of dilute matrix-inclusion composites. Rigid or soft monodisperse spherical inclusions are considered with moderate volume fractions. To conduct this study, Representative Volume Elements (RVE) with an effective local minimum distance between inclusions varying between the sphere’s radius and one-tenth of the radius are built. Numerical finite element calculations on the RVE are performed. The obtained homogenized elastic properties, as well as the phase stress moments (first and second), are compared to Mori–Tanaka estimates, which are well established for this kind of composite. The behavior of local fields (stresses) in the microstructure with respect to inclusion proximity is also analyzed. It follows that the effective properties and phase stress moments converge asymptotically to the Mori–Tanaka estimates when the minimal distance between spheres increases. The asymptote seems to be reached around a distance equal to the sphere’s radius. Effective and phase behaviors show a deviation that can achieve and even exceed (for the second moments) ten percent when the inclusions are close. The impact of the inclusions’ proximities is even more important on local stress fields. The maximum stress values (hydrostatic or equivalent) can be more than twice as high locally. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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19 pages, 6508 KiB

Open AccessArticle

Frictional Energy Dissipation in Partial Slip Contacts of Axisymmetric Power-Law Graded Elastic Solids under Oscillating Tangential Loads: Effect of the Geometry and the In-Depth Grading

by Josefine Wilhayn and Markus Heß

Mathematics 2022, 10(19), 3641; https://doi.org/10.3390/math10193641 - 5 Oct 2022

Cited by 2 | Viewed by 2056

Abstract

Due to the rapid development of additive manufacturing, a growing number of components in mechanical engineering are made of functionally graded materials. Compared to conventional materials, they exhibit improved properties in terms of strength, thermal, wear or corrosion resistance. However, because of the varying material properties, especially the type of in-depth grading of Young’s modulus, the solution of contact problems including the frequently encountered tangential fretting becomes significantly more difficult. The present work is intended to contribute to this context. The partial-slip contact of axisymmetric, power-law graded elastic solids under classical loading by a constant normal force and an oscillating tangential force is investigated both numerically and analytically. For this purpose, a fictitious equivalent contact model in the mathematical space of the Abel transform is used since it simplifies the solution procedure considerably without being an approximation. For different axisymmetric shaped solids and various elastic inhomogeneities (types of in-depth grading), the hysteresis loops are numerically generated and the corresponding dissipated frictional energies per cycle are determined. Moreover, a closed-form analytical solution for the dissipated energy is derived, which is applicable for a breadth class of axisymmetric shapes and elastic inhomogeneities. The famous solution of Mindlin et al. emerges as a special case. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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16 pages, 7264 KiB

Open AccessArticle

Free–Free Beam Resting on Tensionless Elastic Foundation Subjected to Patch Load

by Abubakr E. S. Musa, Madyan A. Al-Shugaa and Amin Al-Fakih

Mathematics 2022, 10(18), 3271; https://doi.org/10.3390/math10183271 - 9 Sep 2022

Cited by 3 | Viewed by 2524

Abstract

Despite the popularity of a completely free beam resting on a tensionless foundation in the construction industry, the existing bending analysis solutions are limited to certain types of loads (mostly point and uniformly distributed loads); these are also quite complex for practicing engineers to handle. To overcome the associated complexity, a simple iterative procedure is developed in this study, which uses the Ritz method for the bending analysis of a free–free beam on a tensionless foundation subjected to a patched load. The Ritz method formulation is first presented with polynomials being used to approximate the beam deflection with unknown constants to be determined through minimization of the potential energy. To account for the tensionless action, the subgrade reaction is set to zero when the deflection is negative. The non-zero subgrade reaction zone is defined by

α_{l} L / 2 < x < α_{r} L / 2

where the coefficients

α_{l}

and

α_{r}

are to be determined iteratively. A numerical example is presented to illustrate the applicability of the proposed procedure for symmetrical and asymmetrical problems. The obtained results show high negative deflection, which proves the occurrence of separation between the beam and the supporting tensionless foundation. This location of negative deflection is called the lifted zone, while the point that separates between the negative and positive deflection is called the lift-off point. A parametric study is then performed to study the effect of the amount of load, stiffness of the beam, and the subgrade reaction on the length of the lifted zone. The results of the parametric study indicate that for the same beam stiffness to subgrade reaction modulus ratio (

E I / k

), the lift-off point remains the same and beams with lower stiffnesses or higher loads deflect more. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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23 pages, 6590 KiB

Open AccessArticle

Pearson Correlation and Discrete Wavelet Transform for Crack Identification in Steel Beams

by Morteza Saadatmorad, Ramazan-Ali Jafari Talookolaei, Mohammad-Hadi Pashaei, Samir Khatir and Magd Abdel Wahab

Mathematics 2022, 10(15), 2689; https://doi.org/10.3390/math10152689 - 29 Jul 2022

Cited by 50 | Viewed by 2800

Abstract

Discrete wavelet transform is a useful means for crack identification of beam structures. However, its accuracy is severely dependent on the selecting mother wavelet and vanishing moments, which raises a significant challenge in practical structural crack identification. In this paper, a novel approach is introduced for structural health monitoring of beams to fix this challenge. The approach is based on the combination of statistical characteristics of vibrational mode shapes of the beam structures and their discrete wavelet transforms. First, this paper suggests using regression statistics between intact and damaged modes to monitor the health of beam structures. Then, it suggests extracting quasi-Pearson-based mode shape index of the beam structures to use them as an original signal in discrete wavelet transforms. Findings show that the proposed approach has several advantages compared with the conventional mode shape signal processing by the discrete wavelet transforms and significantly improves damage detection’s accuracy. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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15 pages, 582 KiB

Open AccessArticle

Calculation of Critical Load of Axially Functionally Graded and Variable Cross-Section Timoshenko Beams by Using Interpolating Matrix Method

by Renyu Ge, Feng Liu, Chao Wang, Liangliang Ma and Jinping Wang

Mathematics 2022, 10(13), 2350; https://doi.org/10.3390/math10132350 - 5 Jul 2022

Cited by 4 | Viewed by 2849

Abstract

In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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21 pages, 1173 KiB

Open AccessArticle

A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model

by Lucille Salha, Jeremy Bleyer, Karam Sab and Joanna Bodgi

Mathematics 2022, 10(10), 1711; https://doi.org/10.3390/math10101711 - 17 May 2022

Cited by 2 | Viewed by 1769

Abstract

Building upon recent works devoted to the development of a stress-based layerwise model for multilayered plates, we explore an alternative finite-element discretization to the conventional displacement-based finite-element method. We rely on a mixed finite-element approach where both stresses and displacements are interpolated. Since conforming stress-based finite-elements ensuring traction continuity are difficult to construct, we consider a hybridization strategy in which traction continuity is relaxed by the introduction of an additional displacement-like Lagrange multiplier defined on the element facets. Such a strategy offers the advantage of uncoupling many degrees of freedom so that static condensation can be performed at the element level, yielding a much smaller final system to solve. Illustrative applications demonstrate that the proposed mixed approach is free from any shear-locking in the thin plate limit and is more accurate than a displacement approach for the same number of degrees of freedom. As a result, this method can be used to capture efficiently strong intra- and inter-laminar stress variations near free-edges or cracks. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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25 pages, 1811 KiB

Open AccessFeature PaperArticle

Condition Number and Clustering-Based Efficiency Improvement of Reduced-Order Solvers for Contact Problems Using Lagrange Multipliers

by Simon Le Berre, Isabelle Ramière, Jules Fauque and David Ryckelynck

Mathematics 2022, 10(9), 1495; https://doi.org/10.3390/math10091495 - 30 Apr 2022

Cited by 6 | Viewed by 5957

Abstract

This paper focuses on reduced-order modeling for contact mechanics problems treated by Lagrange multipliers. The high nonlinearity of the dual solutions lead to poor classical data compression. A hyper-reduction approach based on a reduced integration domain (RID) is considered. The dual reduced basis is the restriction to the RID of the full-order dual basis, which ensures the hyper-reduced model to respect the non-linearity constraints. However, the verification of the solvability condition, associated with the well-posedness of the solution, may induce an extension of the primal reduced basis without guaranteeing accurate dual forces. We highlight the strong link between the condition number of the projected contact rigidity matrix and the precision of the dual reduced solutions. Two efficient strategies of enrichment of the primal POD reduced basis are then introduced. However, for large parametric variation of the contact zone, the reachable dual precision may remain limited. A clustering strategy on the parametric space is then proposed in order to deal with piece-wise low-rank approximations. On each cluster, a local accurate hyper-reduced model is built thanks to the enrichment strategies. The overall solution is then deeply improved while preserving an interesting compression of both primal and dual bases. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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21 pages, 21030 KiB

Open AccessArticle

A Lagrangian DG-Method for Wave Propagation in a Cracked Solid with Frictional Contact Interfaces

by Quriaky Gomez, Benjamin Goument and Ioan R. Ionescu

Mathematics 2022, 10(6), 871; https://doi.org/10.3390/math10060871 - 9 Mar 2022

Cited by 1 | Viewed by 3008

Abstract

We developed a discontinuous Galerkin (DG) numerical scheme for wave propagation in elastic solids with frictional contact interfaces. This type of numerical scheme is useful in investigations of wave propagation in elastic solids with micro-cracks (cracked solid) that involve modeling the damage in brittle materials or architected meta-materials. Only processes with mild loading that do not trigger crack fracture extension or the nucleation of new fractures are considered. The main focus lies on the contact conditions at crack surfaces, including provisions for crack opening and closure and stick-and-slip with Coulomb friction. The proposed numerical algorithm uses the leapfrog scheme for the time discretization and an augmented Lagrangian algorithm to solve the associated non-linear problems. For efficient parallelization, a Galerkin discontinuous method was chosen for the space discretization. The frictional interfaces (micro-cracks), where the numerical flux is obtained by solving non-linear and non-smooth variational problems, concerns only a limited number the degrees of freedom, implying a small additional computational cost compared to classical bulk DG schemes. The numerical method was tested through two model problems with analytical solutions. The proposed Lagrangian approach of the nonlinear interfaces had excellent results (stability and high accuracy) and only required a reasonable additional amount of computational effort. To illustrate the method, we conclude with some numerical simulations on the blast propagation in a cracked material and in a meta-material designed for shock dissipation. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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25 pages, 869 KiB

Open AccessArticle

Nonlinearly Preconditioned FETI Solver for Substructured Formulations of Nonlinear Problems

by Camille Negrello, Pierre Gosselet and Christian Rey

Mathematics 2021, 9(24), 3165; https://doi.org/10.3390/math9243165 - 8 Dec 2021

Cited by 5 | Viewed by 2933

Abstract

We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. We propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach, i.e., nonlinear analogues to the Finite Element Tearing and Interconnecting (FETI) solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. Moreover, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is illustrated on two academic test cases, namely a water diffusion problem and a nonlinear thermal behavior. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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12 pages, 341 KiB

Open AccessArticle

Phi-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers

by Cristiano Maria Verrelli, Cristian Romagnoli, Roxanne Jackson, Ivo Ferretti, Giuseppe Annino and Vincenzo Bonaiuto

Mathematics 2021, 9(13), 1545; https://doi.org/10.3390/math9131545 - 1 Jul 2021

Cited by 5 | Viewed by 4888

Abstract

A harmonically self-similar temporal partition, which turns out to be subtly exhibited by elite swimmers at middle distance pace, is formally defined for one of the most technically advanced swimming strokes—the butterfly. This partition relies on the generalized Fibonacci sequence and the golden [...] Read more.

ϕ

-bonacci butterfly stroke numbers, are proposed to assess such an aforementioned hidden time-harmonic and self-similar structure. An experimental validation on seven international-level swimmers and two national-level swimmers was included. The results of this paper accordingly extend the previous findings in the literature regarding human walking and running at a comfortable speed and front crawl swimming strokes at a middle/long distance pace. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Computational Solid Mechanics)

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