How to Visualize Computational Mechanics: Animating Finite Elements, Continuum Mechanics, and Tensor Calculus

Finite element method has been the dominant technique in computational mechanics in the past decades, and it has made significant contributions to the developments in engineering and science. Nevertheless, finite element method is not well suited to problems having severe mesh distortion owing to extremely large deformations of materials, encountering moving discontinuities such as crack propagation along arbitrary and complex paths, involving considerable meshing and re-meshing in structural optimization problems, or having multidomain of influence in multi-phenomenon physical problems. It is impossible to completely overcome those mesh-related difficulties by a mesh-based method. The highly structured nature of finite element approximations imposes severe penalties in the solutions of those problems. Distinguishing with finite element, finite difference and finite volume methods, meshless method discretizes the continuum body only with a set of nodal points and the approximation is constructed entirely in terms of nodes. There is no need of mesh or elements in this method. It does not posses the mesh related difficulties, eliminates at least part of the FE structure, and provides an approach with more flexibility in the applications in engineering and science. The meshless method started to capture the interest of a broader community of researchers only several years ago, and now it becomes a growing and evolving field. It is showing that this is a very rich area to be explored, and has great promise for many very challenging computational problems. On the one hand, great developments on meshless methods have been achieved. On the other hand, there are many aspects of meshless methods that could be benefit from improvements. A broader community of researchers can bring divergent skills and backgrounds to bear on the task of improving this method. The main objective of this book is to provide a textbook for graduate courses on the computational analysis of continuum and solid mechanics based on meshless (also known as mesh free) methods. It can also be used as a reference book for engineers and scientists who are exploring the physical world through computer simulations. Emphasis of this book is given to the understanding of the physical and mathematical characteristics of the procedures of computational solid mechanics. It naturally brings the essence, advantages and challenging problems of meshless methods into the picture. The subjects in this book cover the fundamentals of continuum mechanics, the integral formulation methods of continuum problems, the basic concepts of finite element methods, and the methodologies, formulations, procedures, and applications of various meshless methods. It also provides general and detailed procedures of meshless analysis on elastostatics, elastodynamics, non-local continuum mechanics and plasticity with a large number of numerical examples. Some basic and important mathematical methods are included in the Appendixes. For the readers who want to gain knowledge through hands-on experience, the meshless programs for elastostatics and elastodynamics are also introduced in the book.

The character of the proper refinement of the elements (mesh) around the boundary is studied. It is shown that it is possible to obtain the highest (optimal) rate of convergence by this refinement.

A computational framework is developed which couples a series of models, each describing vastly different physical events, in order to characterize particle growth (agglomeration) in thermochemically reacting granular flows. The modelling is purposely simplified to expose the dominant mechanisms which control agglomeration. The overall system is comprised of relatively simple coupled submodels describing impact, heat production, bonding and fragmentation, each of which can be replaced by more elaborate descriptions, if and when they are available. Inverse problems, solved with a genetic algorithm, are then constructed to ascertain system parameters which maximize agglomeration likelihood within a range of admissible data.

In computational mechanics, molecular dynamics (MD) and finite element (FE) analysis are well developed and most popular on microscale and macroscale analysis, respectively. MD can very well simulate the atomistic behavior, but cannot simulate macroscale length and time due to computational limits. FE can very well simulate continuum mechanics (CM) problems, but has the limitation on the atomistic level degree of freedom. Multiscale modeling is an expedient methodology with a potential to connect different levels of modeling such as quantum mechanics, molecular dynamics, and continuum mechanics. Developing nanotechnologies require a new simulation method which has both advantages of MD and FE. This paper proposes a new multiscale modeling technique to couple MD with FE. The proposed method relies on combining velocities principle. 1D wave propagation example has been used to illustrate the challenges in coupling MD with FE and to verify the proposed velocity combination approach.

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

Kronecker product algebra is widely applied in control theory. However, it does not appear to have been commonly applied to continuum and computational mechanics (CCM). In broad terms the goal of the current investigation is to extend Kronecker product algebra so that it can be broadly applied to CCM. Many CCM quantities, such as the tangent compliance tensor in finite strain plasticity, are very elaborate or difficult to derive when expressed in terms of tensor indicial or conventional matrix notation. However, as shown in the current article, with some extensions Kronecker product algebra can be used to derive compact expressions for such quantities. In the following, Kronecker product algebra is reviewed and there are given several extensions, and applications of the extensions are presented in continuum mechanics, computational mechanics and dynamics. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker operations on block matrices are introduced. Kronecker product algebra is extended to third and fourth order tensors. The tensorial nature of Kronecker products of tensors is established. A compact expression is given for the differential of an isotropic function of a second-order tensor. The extensions are used to derive compact expressions in continuum mechanics, for example the transformation relating the tangent compliance tensor in finite strain plasticity in undeformed to that in deformed coordinates. A compact expression is obtained in the nonlinear finite element method for the tangent stiffness matrix in undeformed coordinates, including the effect of boundary conditions prescribed in the current configuration. The aforementioned differential is used to derive the tangent modulus tensor in hyperelastic materials whose strain energy density is a function of stretch ratios. Finally, block operations are used to derive a simple asymptotic stability criterion for a damped linear mechanical system in which the constituent matrices appear explicitly.

The tire technology related with the computational mechanics is reviewed from the standpoint of yesterday, today, and tomorrow. Yesterday: A finite element method was developed in the 1950s as a tool of computational mechanics. In the tire manufacturers, finite element analysis (FEA) was started applying to a tire analysis in the beginning of 1970s and this was much earlier than the vehicle industry, electric industry, and others. The main reason was that construction and configurations of a tire were so complicated that analytical approach could not solve many problems related with tire mechanics. Since commercial software was not so popular in 1970s, in-house axisymmetric codes were developed for three kinds of application such as stress/strain, heat conduction, and modal analysis. Since FEA could make the stress/strain visible in a tire, the application area was mainly tire durability. Today: combining FEA with optimization techniques, the tire design procedure is drastically changed in side wall shape, tire crown shape, pitch variation, tire pattern, etc. So the computational mechanics becomes an indispensable tool for tire industry. Furthermore, an insight to improve tire performance is obtained from the optimized solution and the new technologies were created from the insight. Then, FEA is applied to various areas such as hydroplaning and snow traction based on the formulation of fluid–tire interaction. Since the computational mechanics enables us to see what we could not see, new tire patterns were developed by seeing the streamline in tire contact area and shear stress in snow in traction.Tomorrow: The computational mechanics will be applied in multidisciplinary areas and nano-scale areas to create new technologies. The environmental subjects will be more important such as rolling resistance, noise and wear.

A multiscale modeling technique for bridging molecular dynamics with finite element method

With the steady development of computer science, machine learning and data science have made significant progress in recent decades. These techniques generally rely on a substantial amount of data samples to extract the abstract mapping hidden within the data. Hence, these technologies have gradually attracted the attention of researchers in the field of computational mechanics. Combining the recent studies of the authors and other researchers, this paper aims to interpret several forms of applications that integrate machine learning and data science with computational mechanics. In the first application, the core algorithm of the convolutional neural network is implemented to solve the linear elastic finite element problem. A standard finite element equation is transformed into an optimization problem in this method. The method is verified by a plane strain linear elastic finite element problem. The method demonstrates promising accuracy by comparing the results obtained by traditional finite element solver. However, some limitations of this method need to be addressed. First, though the optimization process can be accelerated by GPU, the efficiency of the proposed method is still lower than most mainstream numerical solvers. And, the framework of convolutional neural networks requires that the input layer data should be a constant matrix. This is a major challenge for solving nonlinear finite element equations whose stiffness matrices contain variables. These are the issues worth considerations in future studies. In the second application, a method is proposed to establish the implicit mapping between the effective mechanical property and the mesoscale structure of heterogeneous materials. Shale is employed in this paper as an example to illustrate the method. At the mesoscale, a shale sample is a complex heterogeneous composite that consists of multiple mineral constituents. The mechanical properties of each mineral constituent vary significantly, and mineral constituents are distributed in an utterly random manner within shale samples. Large quantities of shale samples are generated based on mesoscale scanning electron microscopy images using a stochastic reconstruction algorithm. Image processing techniques are employed to transform the shale sample images to finite element models. Finite element analysis is utilized to evaluate the effective mechanical properties of the shale samples. A convolutional neural network is trained based on the images of stochastic shale samples and their effective moduli. The trained network is validated to be able to predict the effective moduli of real shale samples accurately and efficiently. Not limited to shale, the proposed method can be further extended to predict effective mechanical properties of various heterogeneous materials. In the third application, the authors discuss a data-driven computational mechanics framework proposed by Kirchdoerfer and Ortiz. The most outstanding feature of the framework is that explicit material constitutive equations are no longer required. More specifically, experimental material response data are employed in the framework to replace constitutive equations. Combined with traditional compatibility and equilibrium equations, the framework is able to find the optimal stress-strain combination from a material response dataset to best fit the current element. With this framework, the errors and uncertainties induced by the empirical constitutive functions of traditional computational mechanics approaches can be avoided. The aforementioned applications are only the tip of an iceberg in the recent advancement of computational mechanics. Hence, researchers have reasons to believe that there would be more application scenarios that integrate data science and machine learning with computational mechanics in the future. Hopefully, computational mechanics methods with more robustness, efficiency, and fidelity will be developed.

A procedure for extracting equivalent continuum parameters, i.e., (EI) equiv , from skeletal structures has been extended. This procedure which exploits a direct symbolic relationship between the approximation polynomials used in computational mechanics and the equations of continuum mechanics has been used to unify and improve the finite element and finite difference methods. This, in turn, allowed error measures to be put on a solid theoretical foundation and has led to the development of new methods of error identification.

The lower and upper Oldroyd rates are two wellknown objective noncorotational rates. A remarkable property of these rates is that they establish direct relationships between finite strain measures ...

7R3. Me´thode des E´le´ments Finis en Me´canique des Structures. Finite Element Methods in the Mechanics of Structures. (French). - T Gmur. Presses Polytech et Univ Romandes, Lausanne. 2000. 252 pp. ISBN 2-88074-461-X. Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).This textbook deals with the study of some elementary basic problems raised by Applied Mechanics. The book is intended for a large audience: students in Applied Sciences, researchers, engineers, etc. Let Ω be a smooth bounded domain in RN. The problems studied in the present work are of the following (weak) form: find a vector field u such that ∫Ω vT[Du-f]dΩ=0 ∀v,where D is a certain linear differential operator and f is a given vector field. Throughout the work, the study is limited to the case of linear elliptic partial differential equations of second order. In many cases, the exact solution of this problem (if it exists!) cannot be found explicitly. That is why several methods have been developed in Numerical Analysis in order to approximate the solutions of wide classes of differential equations. The Finite Element Method was introduced in the 50s, and it can be viewed as an extension of the classical Galerkin Method. The Finite Element Method consists in the decomposition of Ω into a finite number of subdomains, called finite elements. The solution of the initial problem is then found as an assemblage of functions with compact support. After a short introduction, the author introduces in Chapter 2 the strong and the weak mathematical formulation in the one-dimensional case. This elementary frame-work enables the author to explain carefully how numerical methods can be applied for approximating the solution. The Finite Element Method is based in this simple case by an integral approach. In the next chapter, the weak formulation of the approached problem is developed, and one-dimensional linear finite problems of second degree are considered. Chapter 4 is devoted to the presentation of the weak formulation to linear bidimensional problems of the second degree. The study is developed on the model of the heat-transfer equation. In this case, the approach is based on planar finite elements. The author develops the mathematical treatment of the problem, which includes convergence and numerical integration. A general situation is treated in Chapter 5, which includes applications in Linear Elasticity for two or three dimensions. The last chapter is devoted to some applications of the Finite Element Method. Here, the numerical solutions are confronted to experimental results. In this chapter, but also throughout the book, the exposition simplifies many results that have so far only been accessible in several journal articles. In short, there is an enormous wealth of content provided by the author, much of it cannot be found in any single source. Certainly, many techniques related to the Finite Element Method are discussed, so the title is unquestionably appropriate. The book and its extensive bibliography (100 titles strictly related to the subject) should serve as a tremendous reference for all researchers in the field and certainly belongs on the bookshelf next to other references on the Finite Element Method. One of the reasons why the author is able to cover such vast material in the book is that he tends to develop the main ideas exactly to the right degree of generality that he needs. Moreover, the treatment is always done by progressing from the special to the general case, essentially avoiding pedantic verifications. Sometimes the proof is even given through the right picture. In this case, the author has really saved the reader from a lot of unnecessary heavy notation. Moreover, the notation is not only simple, but also everywhere consistent. One basic question this reviewer would like to answer is whether this book is meant for students. My feeling is that, first of all, it is very nice for students to see so many concrete examples and pictures. The reader here is well motivated by simple, but illuminating examples before being faced with general notions. This is due quite systematically in the book, where examples and pictures always precede the proofs and allow those to be easily stated and easily understood. This reviewer concludes that Me´thode des E´le´ments Finis en Me´canique des Structures is an attractive book, full of concrete information, which gives a clear and lucid view of the current knowledge on the resolution of differential equations with the Finite Element Method.

Objective To establish a three-dimensional (3D) finite element model of a swine mandible, simulate the dynamic procedure of bullet damaging the swine mandible and explore a finite ele-ment analysis method on maxiilofacial gunshot wound. Methods The digital imaging and communica-tions in medicine (DICOM) data obtained from CT scanning of a swine mandible were remerged into a 3D finite element model of the original specimen through Mimics and ANSA software, then a simulation of 3D finite element model penetrated by a 7.62 mm bullet was carried out through LS-DYNA software. The simulation data were compared with those from animal experiment in laboratory to test the feasibility of 3D finite element model and the simulation method. Results A 3D finite element model of a swine mandi-ble was established, with highly identical geometric size with the specimen. In the meantime, the dynam-ic process of a 7.62 mm bullet damage to the model was successfully simulated. Data from the simulation and those from animal experiment showed a high level of consistency. Conclusion 3D finite element method is prosperous in application in basic research on maxillofacial gunshot wound. Key words: Mandible; Wounds, gunshot; Finite element analysis, three dimensional; Bioballistics

9R74. Cardiovascular Solid Mechanics: Cells, Tissues, and Organs. - JD Humphrey (Dept of Biomed Eng, Texas A&M Univ, College Station TX 77843-3120). Springer-Verlag, New York. 2002. 757 pp. ISBN 0-387-95168-7. $99.00.Reviewed by M Epstein (Dept of Mech Eng, Univ of Calgary, 2500 University Dr NW, Calgary AB, T2N 1N4, Canada).Ambitious both in scope and depth, this book constitutes a remarkable achievement. Predicated on the principle that the next generation of biomechanicists should be as proficient in Continuum Mechanics as in Biology and in formulating simple models from raw experiments, this 750-page book attempts to encompass the necessary combined background in one volume suitable for use as a text. It is divided into three parts. The first part, occupying roughly a third of the book and entitled Foundations, can be considered as an almost stand-alone textbook in Continuum Mechanics and the Finite Element Method. Keeping in mind the intended application to soft tissues, the treatment emphasizes geometrical and material nonlinearities. It is doubtful that students without any previous background in either Continuum Mechanics or Finite Elements might be able to acquire a working knowledge in either subject from this book alone. On the other hand, students already having an introductory exposure to these subjects will be able to see the whole picture and will profit enormously from the relatively high-level and concise style of the presentation. The theoretical treatment is supplemented with a chapter on experimental methods. If the book were to be used as text, a two-semester format would be ideal, with the first semester entirely devoted to the study of the first part of the book so as to provide the students with a sound foundation in theoretical, numerical, and experimental methods. Taking into consideration the aforementioned principle, the effort will not be wasted. The second, and largest, part of the book is dedicated to Vascular Mechanics. A good description of the histology and physiology of the arterial wall is followed by material considerations, such as symmetry, inhomogeneity, incompressibility, and residual stress. It is here where the knowledge gained in the foundations becomes important, since the general experimental observations are implemented within a consistent constitutive framework. Perhaps the most interesting chapter of this second part of the book is the one devoted to Vascular Disorders (hypertension, aneurisms, arteriosclerosis). Vascular adaptation is given a separate chapter in which modern theories of kinematic growth are discussed among other ideas. The third, and final, part of the book is devoted to Cardiac Mechanics. It consists of a lone 120-page chapter on the normal mature heart. The lack of even a short chapter on cardiac disorders is noted. Each chapter is followed by a set of challenging exercises ranging from historical reviews to detailed calculations. A wealth of references is proof that Cardiovascular Solid Mechanics: Cells, Tissues, and Organs is not just intended as a text, but also as a valuable reference for researchers in soft-tissue mechanics. It should be purchased by libraries for general use and by individuals that would like to have an excellent, handy, comprehensive, and useful reference book on their shelves.

Increasing the efficiency of computational welding mechanics by combining solid and shell elements