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This text presents a highly original treatment of the fundamentals of FEM, developed using computer algebra, based on undergraduate-level engineering mathematics and the mechanics of solids. The book is divided into two distinct parts of nine chapters and seven appendices. The first chapter reviews the energy concepts in structural mechanics with bar problems, which is continued in the next chapter for truss analysis using Mathematica programs. The Courant and Clough triangular elements for scalar potentials and linear elasticity are covered in chapters three and four, followed by four-node elements. Chapters five and six describe Taig’s isoparametric interpolants and Iron’s patch test. Rayleigh vector modes, which satisfy point-wise equilibrium, are elaborated on in chapter seven along with successful patch tests in the physical (x,y) Cartesian frame. Chapter eight explains point-wise incompressibility and employs (Moore-Penrose) inversion of rectangular matrices. The final chapter analyzes patch-tests in all directions and introduces five-node elements for linear stresses. Curved boundaries and higher order stresses are addressed in closed algebraic form. Appendices give a short introduction to Mathematica, followed by truss analysis using symbolic codes that could be used in all FEM problems to assemble element matrices and solve for all unknowns. All Mathematica codes for theoretical formulations and graphics are included with extensive numerical examples.

### Chapter 1. Finite Element Basics with the Bar Element: Uniaxial Deformations—Interpolants, Stiffness Matrices and Nodal Loads

In this introductory chapter, we emphasize important concepts of the finite element method using simple, intuitive examples. An undergraduate engineering mathematics background should be adequate. However, the basic operations with linear algebra and differential equations are reviewed within the context.
The finite element method, even for unbounded media, projects continuum solutions—governed by (partial) differential equations—into a finite dimensional vector space. Strikingly enough, the merit of the method permits the introduction of all basic (physical and mathematical) ideas with one-dimensional bar examples. The deformation analysis of a system of bars captures all essential aspects of thermo-mechanical behavior. Thus, this chapter provides a foundation for the topics developed in this textbook. In association with characteristic “internal forces,” which guarantee equilibrium to yield quality solutions, there are independent Rayleigh displacement modes. These finite number of basis functions (blending functions or interpolants) are the fundamental objects of the finite element method. The resulting nodal forces and displacements yield symmetric (positive semi-definite) system matrices.
The “energy minimization” concept is introduced using a single spring element (a single degree-of-freedom system). In order to reinforce the idea of degrees-of-freedom and of the energy-like scalars, the physical Rayleigh mode is introduced as the fundamental pattern of deformation.
Generalization, e.g. frame invariance concepts, in two- and three-dimensions, involves “inversion” of rectangular matrices; the associated pseudoinverse concept is introduced within that context. Discrete representation with indicial notation is described in detail, and weak solutions are introduced within a smaller dimensional vector space.
Gautam Dasgupta

### Chapter 2. Truss Analysis: A Structural Mechanics Tour of the FEM—Nodal Equilibrium and Compatibility

Linear elastic behavior and small displacements and rotations (Spencer, Continuum mechanics. Longman, London, 1980) are assumed throughout. This model of linear systems allows us to obtain the nodal forces in terms of the bar stiffness matrices and nodal displacements that are the primary variables. The equilibrium equations can then be constructed in terms of displacements at nodes. The given forces and displacements are assumed to be prescribed as nodal quantities. The resulting system equation, which is in the matrix form, can be obtained in terms of nodal displacements and solved by employing the Mathematica function Solve.
A very important area of numerical analysis, i.e., solving positive definite simultaneous equation systems, will not be addressed in this textbook.
This chapter introduces Mathematica codes. To get started, a summary introduction is provided in Appendix A that should be studied before reading the current chapter.
After the first reading, Appendix B should be reviewed for additional examples and theoretical analysis.
Gautam Dasgupta

### Chapter 3. Courant’s Triangular Elements with Linear Interpolants

For one-dimensional domain the Lagrange interpolation, with the polynomial structure, furnishes an excellent approximation. For two non-zero data points, the Lagrange interpolant is a straight line. Piecewise parabolic approximations are also quite popular.
The linear interpolation over a triangulated mesh is the two-dimensional counterpart of the Lagrange interpolation with two data points.
Courant illustrated two-dimensional spatial discretization for the structural mechanics problem of torsion pertaining to a non-circular prismatic bar (Courant, Bull Am Math Soc 49(1):1–29, 1943) (In his first sentence of the last paragraph, Courant wrote: “Of course, one must not expect good local results from a method using so few elements.” Obviously, he was talking about a finite number of elements. We can therefore credit Courant to be initiator of finite elements.). He demonstrated the intimate connection between triangulation and linear interpolants.
A collection of scholarly papers appears in Michal Křížek and Stenberg (eds) (Finite element methods: fifty years of the Courant element. CRC Press, Boca Raton, 1994 (Marcel Dekker, Jyvaskyla, 1993)). This chapter prepares readers to undertake such in-depth studies.
Gautam Dasgupta

### Chapter 4. Clough’s Plane Triangular Elements for Linear Elastic Problems: Virtual Work Substituting Variational Principle

Clough (Proceedings, 2nd conference on electronic computation, A.S.C.E. structural division, Pittsburgh, PA, pp 345–378, 1960) introduced the term finite element with his classical displacement formulation of plane strain triangles. These are plane elements with constant stress distributions. Analogous plane stress triangular elements can be similarly formulated. The corresponding (The triangle is the simplest polygon in $$\mathfrak{R}^{2}$$ and the tetrahedron is its three-dimensional counterpart.) three-dimensional cases of tetrahedral elements do not pose any conceptual difficulty. To emphasize this natural extension, we first review the three-dimensional field equations of continuum mechanics, and then formulate the element stiffness matrix for triangular domains.
An important feature of triangular elements is that their shape functions are linear polynomials in the physical (x, y) coordinate variables. This renders the stress and strain fields to be constant within an element. Hence, the point-wise equilibrium is always satisfied unconditionally. Thus, it does not matter even if the linear elasticity formulations are coupled vector field problems, the shape function vectors with one zero component still qualify to be admissible functions. (Courant (Bull Am Math Soc 49(1):1–29, 1943) emphasized this concept starting from his Sect. II as he focused on the Rayleigh–Ritz Method, just above his equation number (12).) Hence the “uncoupling (This is elaborated for four-node elements in Sect. 5.​4.)” of the shape function vectors, à la Courant scalar field problems (Courant, Bull Am Math Soc 49(1):1–29, 1943), still persists even for coupled vector field problems of elasticity (It will be stated in Eqs. (5.​7) and (5.​8) that the independence of displacement vectors under a single nodal displacement is standard even for four-node elements that violate point-wise equilibrium.) when spatial discretization with triangles is invoked. (Similar conclusions can be drawn when trapezoidal elements are used for three-dimensional linear elasticity problems.) Assuming the other parts of a finite element computer program to be without any flaw, any linear stress field on arbitrary domains will be exactly reproduced irrespective of meshing details. (In Chap. 6 we analyze this concept (the patch test) in depth.) From the theoretical standpoints, this observation raises a valid question as to whether other elements with more nodes will have such a property that brings the finite element method close to very reliable approximation of problems with boundaries of arbitrary shapes.
Clough employed the physical concepts of virtual work to identify the entries of stiffness matrices and nodal forces to be virtual work quantities. The variational principle (though mathematically elegant) was not essential because, unlike Courant, Clough was not addressing the abstract class of elliptic boundary value problems for which the “rigid” and “natural” boundary conditions always emerge from the associated variational principle. It is important to state that for physical problems, where there is a notion of energy, (due to the self-adjointness property of the partial differential equations of mathematical physics) the principle of virtual work is congruent with the variational formulation of Ritz (vide Ritz, J Reine Angew Math 135:1–61, 1908; Gaul, In: GAMM: Proceedings in applied mathematics and mechanics. Springer, Rotterdam, 2011).
Gautam Dasgupta

### Chapter 5. Taig’s Quadrilateral Elements

A Clarification Taig did not use the term isoparametric,in his 1962 report (Taig, Structural analysis by the matrix displacement method. Tech.rep., British Aircraft Corporation, Warton Aerodrome: English Electric Aviation Limited, Report Number SO 17 based on work performed ca. 1957, 1962). This seminal publication introduced his quadrilateral elements. In 1968,Ergatoudis, Irons, and Zienkiewicz coined (To the best of the author’s knowledge, the use of the term isoparametric thus ensued.):“…for lack of a better name,as the isoparametric quadrilateral.”  (Ergatoudis et al., Int J Solids Struct 4(1):31–42, 1968).In 1974, Ian C. Taig and Bruce Irons were awarded the Von Karman prize, for the introduction of isoparametric element concepts. From §3.3, page 167 of Strang and Fix (An analysis of the finite element method. Prentice-Hall, Inc., Englewood Cliffs, 1973) (Strang and Fix do not credit Taig in their monograph.):“isoparametric means that the same polynomial elements are chosen for coordinate changes for the trial functions themselves;…”
Computation vis-à-vis Calculation To distinguish calculation from computation, we observe that Taig’s interpolants along with numerical quadratures, Li et al. provide a calculation method whereas a general implementation on all quadrilaterals, including convex ones with curved boundary establishes a computational procedure (Li et al., Comput Methods Appl Mech Eng 197(51):4531–4548, 2008).
Triangular to Quadrilateral Finite Elements Ian Taig, in his 1962 report, introduced four-node elements capturing stress/strain profiles beyond Clough’s triangles that are restricted to constant stress/strain distributions. Taig’s intent was to capture bending stresses (spatially linear axial stresses/strains) in beam elements that Taig termed panels. He guaranteed linear displacement on straight boundary edges. This ensured perfect inter-element compatibility. Wilson interpreted Taig’s interpolants as the elements’ natural coordinates (Wilson, Static and dynamic analysis of structures. Computers and Structures, Inc., Berkeley, 2003). Subsequently, researchers analyzed bending in plates and shells, e.g. Zhu et al. (Int J Mod Phys B 19(01n03):687–690, 2005), by overcoming thetoo-stiffconstant stress/strain elements.
Taig’s Interpolations in the Physical (x, y)-FrameAll textbooks and industrial manuals extensively cover the plane quadrilateral elements that Taig introduced.Research and teaching materials, which can be easily accessed on the internet, need no repetition. This chapter analyzes Taig’s formulation in algebraic terms in the interest of brevity and clarity.We can derive that, in general, Taig’s parametric interpolants involve the square root of quadratic expressions in the physical (x, y) frame.For trapezoids Taig’s interpolants become rational polynomials—a quadratic in (x, y) divided by a linear expression in (x, y).Wachspress addressed this issue completely (Wachspress, A rational finite element basis. Academic, New York, 1975; Rational bases and generalized barycentrics: applications to finite elements and graphics. Springer, New York, 2015). Convex, concave, and elements with curved boundaries are treated using projective geometry concepts.
Symbolic Closed-Form ExpressionsTaig in §2.3 of his 1962 report formulated closed-form expressions of element stiffness matrices for a rectangular element that he termed “Rectangular Sheet panel.” He revisited “Triangular Sheet panel”  in §2.4 of his 1962 report and furnished closed-formexpressions for the stiffness matrix. This encouraged the author to analyze all quadrilateral elements using symbolic computational tools.
Taig’s Analysis of Trapezoidal and General Quadrilateral ElementsTaig clearly pointed out the difficulties in determining the element stiffness matrix for non-rectangular elements.He presented stiffness matrices for trapezoidal elements in closed-form (vide pages 59 and 60 of Taig (1962)—“MATRIX IV”). General convex or concave quadrilateral elements were not included in the Report.
Numerical EvaluationTaig, in §6 of his 1962 report,extensively described the steps employed in “Formulation for automatic Computation.” Programs for DEUCE computers (of 1959) were elaborated.
Impact of Taig’s WorkFollowing the spirit of the aforementioned pioneering work of Ian C. Taig, complete Mathematica codes are furnished in this textbook for scalar field problems.However,generalization to deal with elastic elements, where the displacement vector components are coupled through Poisson’s ratio, demanded looking back to Lord Rayleigh’s formulations.
Gautam Dasgupta

### Chapter 6. Irons’ Patch Test

To judge the quality of any element formulation, starting from resolving controversies that surrounded Taig’s four-node element, the patch test (The author failed to locate the first publication where Irons introduced this revolutionary concept of the patch test. In internal documents of Rolls-Royce, Derby, UK, most likely in the early sixties, definitely after the groundwork of Taig (Structural analysis by the matrix displacement method. Tech. rep., British Aircraft Corporation, Warton Aerodrome: English Electric Aviation Limited, Report Number SO 17 based on work performed ca. 1957, 1962), Irons presented the concept. In 1965 (Irons et al., Triangular elements in plate bending–conforming and nonconforming solutions. In: Proceedings of the conference of matrix methods. Wright-Patterson Air Force Base, Ohio, 1965), Irons first authored a paper on plate bending where the issue beyond constant stress elements was unavoidable.) furnished a guideline. Irons thus made the most fundamental contribution in the science of finite elements. Irons’ (co-authored) books, e.g. Irons and Razzaque (Experience with the patch test for convergence of finite elements method. In: Aziz A (ed) Mathematical foundations of the finite element method with application to partial differential equations. Academic, New York, pp 557–587, 1972), Irons and Ahmad (Techniques of finite elements. Ellis Horwood, Chichester, 1980), Irons and Shrive (Finite element primer. Wiley, New York, 1983), bear testament to his tremendous insight in analyzing spatially discretized field problems of mathematical physics. This innovative notion has inspired researchers, for example Anufariev et al. (Exactly equilibrated fields, can they be efficiently used in a posteriory error estimation? In: Physics and mathematics. Scientists notes of the Kazan University, vol 148. Kazan University, Kazan, pp 94–143, 2006), Stewart and Hughes (Comput Methods Appl Mech Eng 158(1):1–22, 1998), over the last 50 years or so (Professor Robert L. Taylor in: http://​www.​ce.​berkeley.​edu/​projects/​feap/​example.​pdf furnishes explanations with numerical examples.) and no doubt many more interesting papers will appear on this subject.
Gautam Dasgupta

### Chapter 7. Four-Node “Locking-Free” Elements: Capturing Analytical Stresses in Pure Bending: For Two Orthogonal Directions

We can exactly reproduce linear axial stresses without shear (To satisfy equilibrium point-wise, the (u, v) components of a typical displacement vector must be coupled through the Poisson’s ratio ν. ), in four-node (plane) elements. Neither their geometrical shapes—including concavity—nor the Poisson’s ratio ν pose any impediment. This locking-free condition can only be achieved when the neutral axis, under pure bending, is aligned along one of the two pre-selected directions.
Here, an element shape function, which consists of two quadratic polynomials in (x, y), is a linear combination of Rayleigh mode vectors that satisfy point-wise equilibrium. The unique coefficients for each shape function involve the element nodal coordinates and the Poisson’s ratio.
Following Clough’s guidelines, (Clough, Comput Struct 12:361–370, 1980) especially the Quotation III of the Introduction, the entries in the element stiffness matrix and the (equivalent) nodal forces (to substitute for a given spatial profile of the boundary traction) are determined as virtual work quantities (Note that for structural mechanics problems, the virtual work principle is absolutely equivalent to the Ritz variational formulation (Ritz, J Reine Angew Math 135:1–61, 1908; Wendroff, Math Comput 19(90):218–224, 1965), making the notion of variational crimes (Strang, Variational crimes in the finite element method. In: Aziz AK (ed) Mathematical foundations of the finite element method with application to partial differential equations. Proceedings Symposium, University of Maryland, Baltimore. Academic, New York, pp 689–710, 1972; Gander and Wanner, SIAM Rev 54(4), 2012) completely irrelevant here.) (by exact integration).
This Rayleigh modal approach models:
(1)
compressible media (The conventional finite element method with nodal degrees-of-freedom prevails.), with the Poisson’s ratio in the range − 1 < ν < 1∕2

(2)
and in Chap. 8, incompressible media (Where the eight nodal displacements cannot be arbitrarily prescribed.), with ν = 1∕2

In pure bending, which is associated with an arbitrarily oriented neutral axis, let us reiterate that quadrilateral plane elements, with eight degrees-of-freedom, will invariably yield non-zero shear stresses. There is not an adequate number of degrees-of-freedom to guarantee point-wise equilibrium with shear-free linear stresses. In that sense, the element cannot be entirely defect-free (MacNeal, Finite Elem Anal Des 5(1):31–37, 1989). This problem is elaborated in Chap. 9
Gautam Dasgupta

### Chapter 8. Incompressible Plane Strain Elements: Locking-Free in the x and y Directions

A procedure, which takes the Poisson’s ratio ν to be exactly 1∕2, is developed here by selecting only those Rayleigh mode vectors that point-wise satisfy the isochoric, i.e., zero-volume-change, condition. A more research-type exposé can be found in Dasgupta (Acta Mech 223:1645–1656, 2012). A continuum mechanics treatment of the isochoric formulation in Cartesian coordinates can be found in Spencer (Continuum mechanics. Longman, Harlow, 1980) (Spencer indicated the dilatation by Δ; we use Θ instead, in this textbook.).
For four-node plane strain incompressible elements, the Rayleigh polynomial vectors, which satisfy equilibrium point-wise, are associated with:
1.
three rigid body modes, which trivially satisfy incompressibility

2.
uniform deviatoric stresses (2 modes), and these necessarily conform with the incompressibility condition

3.
two incompressible linearly varying axial strains without shear (ε xx + ε yy = 0, ε xy = 0; ε xx , ε yy : linear combinations of (x, y).).

Within each element level, a uniform element pressure must be taken as the eighth independent variable. Interestingly, the pseudoinversion (in Sect. 1.​8, the Moore–Penrose weak inverse of rectangular matrices is described) of the modal-to-nodal (eight rows by seven columns) rectangular matrix makes this chapter very distinct from popular isochoric formulations.
Incompressibility disqualifies the nodal displacements from being counted as the degrees-of-freedom. An independent nodal displacement invariably alters the element volume. Hence, the equilibrium and nodal compatibility equations are solved by determining the weights of the seven Rayleigh mode vectors and one (constant) pressure variable per element (To facilitate symbolic programming an additional notation to indicate the element number, which is encased within superscript parentheses, has been introduced.). Thus, without assembling the global stiffness matrix, all nodal displacements as well as all unknown element pressures are determined. As usual, concavity does not pose any difficulty.
Gautam Dasgupta

### Chapter 9. Conclusions—Plane Elements: Polynomial Stresses of Degree n

The patch test, which was one of Iron’s brilliant ideas, initiated various innovations in formulating new finite elements. In the spirit of eliminating MacNeal’s (Finite Elem Anal Des 5(1):31–37, 1989) element deficiencies or “failure modes” all terms from the Rayleigh modes, which are necessarily in equilibrium, are included. Here the minimum number of additional nodes, where equilibrium of forces and displacement compatibility should be enforced, are preferred over rotational degrees-of-freedom.
The Poisson’s Ratio: Its Rôle in Dimensional Analysis
This parameter couples the displacement components that are the primary variables in finite elements discussed here. Accuracy, in the algorithmic sense, cannot be attained unless this non-dimensional quantity is explicitly inserted into the test functions that indeed are the Courant coordinate functions in the Ritz variational principle.
The equilibrated Rayleigh modes, for elasticity problems, will invariably contain the Poisson’s ratio.
Continua Abstracted to a Collection of Distinct Points
The possible mismatch of tractions and displacement fields on element boundaries is not considered. Clough’s systematic development with displacement test functions demands equilibrium and compatibility only at the system nodes. This complies with all the requirements for the virtual work principle to be applicable. The needs posed by the Ritz and Courant variational formulation for elliptic boundary value problems are met as well.
For a prescribed error bound, the allowable element sizes, i.e., the fineness of discretization, must be modeled according to the h-convergence criterion. In practice this is achieved by trial and error.
Limitations
Error analysis for coupled vector field elliptic partial differential equations is highly technical and quite rare. However, a considerable body of research exists for scalar elliptic boundary value problems. These are appropriate for temperature distributions but not when the displacement vector field is a partial sum mentioned in the Ritz variational formulation (Ritz, J Reine Angew Math 135:1–61, 1908). This topic is outside the scope of this textbook.
Locking-Free Four Node Elements
The patch test was primarily motivated to account for bending that could not be accommodated within constant stain/stress triangular finite elements. In four node plane elements, pure bending stresses cannot be reproduced in all directions simultaneously. We show that we need at least five nodes, i.e., ten degrees-of-freedom, to reproduce pure bending stresses pertaining to neutral axes in arbitrary directions. Thus, in Dasgupta (Acta Mech 223(8):1645–1656, 2012) patch tests for the four-node element was restricted to two-orthogonal pure bending cases.
Generalization to Capture Higher Degree Stress Fields
Following Clough’s classical displacement formulation, higher degree stresses demand more element nodes. In order to enhance computation from n − 1 to n,  n = 1, 2⋯ degrees of stress distributions, we need at least 2 additional nodes for plane elasticity problems (and 2n + 3 additional ones for solid three-dimensional elements).
Symbolic Computation
Mathematica helps to carry out exact integration that avoids subjectivity in selecting numerical quadratures to integrate strain energy density functions within the element domain.
Rayleigh Mode Formulation and Incompatible Elements
Here, the finite element basis functions are linear combinations of Rayleigh mode vectors that satisfy point-wise equilibrium. The displacements along the boundary will not be linear beyond constant strain elements. This treatment debuts in this textbook. Hence, a summary of the core ideas is in order here.
Concluding Remarks
The intent of this textbook has been to develop the finite element concepts following Ritz’s foot steps. He introduced a series of polynomial test functions and calculated their weights by minimizing an energy-type integral. Courant harmonized spatial discretization with piecewise continuous functions. Iron’s patch tests safeguards against variational crimes. Symbolic computation permitted treating results in the unadulterated algebraic form. Using object-oriented procedural programming languages, modern computers make large scale design-analysis a reality.
Gautam Dasgupta