A Trefftz collocation method (TCM) for three-dimensional linear elasticity by using the Papkovich-Neuber solutions with cylindrical harmonics
Guannan Wang a, Leiting Dong b,*, Satya N. Atluri a
a Center for Advanced Research in the Engineering Sciences, Texas Tech University, Lubbock, TX 79409, USA
b School of Aeronautic Science and Engineering, Beihang University, Beijing, China
Abstract A Trefftz collocation method (TCM) is proposed for solving three-dimensional (3D) linear-elastic boundary value problems. By using the Papkovich-Neuber (P-N) general solutions, Trefftz trial functions are expressed in terms of cylindrical harmonics. Both non-singular and singular harmonic functions are included, facilitating the study of interior and exterior problems. To mitigate the problem of ill-conditioning, two steps are adopted: The first step is to introduce a characteristic length of the domains of interests into the Laplace equation, and the second step is to scale each column of the coefficient matrix in the established system of linear equations using another multi-scale characteristic length, letting each column have the equal norms. Several examples are presented to validate the proposed 3D Trefftz collocation method. The completeness of the trial functions, the effect of the scaling techniques, and the accuracy of solutions are also discussed.
Keywords: Trefftz method; 3D elasticity; Papkovich-Neuber solution; cylindrical harmonics; ill-conditioning
1. Introduction
Trefftz method [1] is a meshless method for the numerical solutions of boundary value problems, where the trial functions are represented as a linear combination of functions that satisfy the governing differential equations a priori. Trefftz method can be categorized as direct and indirect formulations, depending on whether Trefftz basis functions are applied as test or trial functions [2]. The unknown coefficients in the Trefftz functions are usually determined by matching the boundary conditions. It has also been found that the collocation technique is the one of the simplest methods in dealing with boundary conditions and providing accurate numerical results [3-4]. Trefftz method has been successfully applied in numerous physical, mathematical and engineering situations, including Helmohotz equation [5-7], Poisson equation [8], Laplace equations [9-11], eigenvalue problems [7], elasticity problems [12], plate bending problems [13], free vibration problems [6]. Trefftz method was also associated with finite element (FE) in generating hybrid-Trefftz (HT) FE model since the early contributions by Jirousek and his coworkers [14-15]. Dong and Atluri applied Trefftz Voronoi Cell Finite elements in micromechanical modeling of heterogeneous materials with different shaped inclusions or voids [16-17], in which the complete trial functions predicted accurate effective properties and localized stress concentrations. A detailed review work of similar topics was conducted by Qin [18]. The 3D elasticity solutions have been studied for decades. To the best knowledge of the authors, the solutions proposed by [19-20] are the most complete solutions ever developed based on cylindrical coordinates until the present stage. However, these solutions only focused on the non-singular terms while ignoring the terms that cause singularity at the origins of the coordinates, which limits their usefullness for exterior problems. Based on the situation described above, along with the fact that the 3D elasticity P-N solutions using spherical harmonics and elliptical harmonics were developed by [16-17] for the purpose of solving spherical/elliptical shaped subject domains, we develop a complete 3D Trefftz method by employing P-N solutions that automatically satisfy Navier’s equations in terms of the cylindrical harmonics, including both non-singular and singular terms, in order to deal with not only interior but also exterior problems. The boundary conditions are applied through collocation techniques, where the P-N solutions are matched with the displacement or traction boundary conditions point by point. Despite of its wide applications, traditional Trefftz method still suffers from severe ill conditioning problems after establishing system of equations in certain scenarios. Liu [21-22] resolved the problem for Trefftz solutions of Laplace equation by introducing characteristic lengths and significantly reduced the condition numbers. Here similar procedures are adopted to rescale the cylindrical harmonic terms and the systems of equations, where two characteristic lengths are introduced. Finally, several elasticity examples within different domains of interests are presented to prove the accuracy and stability of the present method. The organization of the present work follows as: Section 2 exhibits the governing equations for 3D elasticity, and presents the P-N solutions in which Laplace potential is solved using cylindrical harmonics. Since unscaled functions of cylindrical harmonics include several types of functions, which will cause severely ill-conditioned results, two scaling techniques are introduced in Section 3 to alleviate the badly scaled conditions. Section 4 presents several numerical examples, proving the accuracy of Trefftz solutions and collocation technique, as well as the necessity of using scaling techniques. Section 5 concludes this presentation.
2. Trefftz trial functions for 3D linear elasticity
2.1 The Papkovich-Neuber solution
The general solutions of 3D linear elasticity should satisfy the equations of stress equilibrium, strain-displacement gradient compatibility, as well as constitutive relations:
∇⋅σ +f =0 (1)
1 ∗ ε =[ ∇uu +∇ ( )] (2) 2 σ =λµtr()ε I + 2 ε (3)
where σ , ε , and u are stresses, strains, and displacements, and
T T T σ = [,,,,,]σσσσσσxx yy zz xy yz xz , ε = [,,,,,]εεεεεεxx yy zz xy yz xz and u = [,,]uuuxyz . ∇⋅ and
∇ refer to the divergence and gradient, respectively. f is the body force vector acting on elastic body; ()∇u ∗ is the transpose of ∇u ; I is the identity tensor; λ and µ are
Lame’s constants given as λν=E (1 +− ν ) (1 2 ν ) and µν=E 2 (1 + ) for isotropic materials. Navier’s equations can be deduced from Eqs. (1-3), representing equilibrium equations in terms of displacement components [20,23]: (λµ+ )( ∇ ∇⋅u ) + µ ∇2 uf + = 0 (4)
where ∇2 is the Laplacian operator. The difficulty of solving the 3D Navier’s equations arises from the coupled displacement components appearing in each of the equations in Eq. (4). Papkovich [24] and Neuber [25] indicated that the general solutions can be expressed in terms of harmonic functions if the body force is neglected:
u=4(1 −v ) B −∇ ( RB ⋅ + B0 ) (5)
in which B and B0 are vector and scalar harmonic functions (P-N potentials), and
Re= xi i denotes the position vector. It can be easily noticed that the equilibrium equations, Eq. (4), only contain three unknown displacement components in Cartesian coordinate. Therefore, it is acceptable that only the vector harmonic function B
(including three components) is kept, and dropping the scalar function B0 leads to a new solution: u=4(1 −v ) B −∇ RB ⋅ (6) which is a complete solution for the infinite domain. However, one can prove that the general solution of Eq. (4) can be solved as Eq. (6) only under the condition ν ≠ 0.25 for a simply-connected region. Later on M.G. Slobodyansky proved that for a simply connected domain, a complete general solution can be obtained by expressing B0 as a function of B [23], leading to: u=4(1 −v ) B + R ⋅∇ B −∇ RB ⋅ (7)
2.2 Cylindrical Harmonics as Papkovich-Neuber potentials
Consider a position vector within Cartesian coordinates [,xxx123 , ] and the corresponding
cylindrical coordinates [qqq123 , , ]= [,,] Rθ z, they have the relation:
xR1=cosθθ , xR 23 = sin , xz = (8)
The base vectors e12= eerz,, = eeθ 3 = e are in the directions of the radius of circles, the tangents to circles and the axis of the concentric cylinders, respectively. Now we have ∂∂xx ∂ x 11==−=cosθθ , R sin , 1 0 ∂∂Rzθ ∂ ∂∂xx ∂ x 22=sinθθ , = R cos , 2= 0 ∂∂Rzθ ∂ ∂∂∂xxx 333=0, = 0, = 1 ∂∂∂Rzθ (9) and ∂qs 1 ∂x ∂∂RR =k ⋅=δ 2 s, rsrsHH r s (10) ∂xks Hq ∂ ∂∂ q q where HH1 =r =1, HHr2 =θ = , HH3 =z =1 are referred to as Lame’s coefficients. The gradient of a scalar field can be expressed as: 1 ∂ψ ψ = = gradaasse , s s . (11) Hqs ∂ Then the Laplace equation in cylindrical coordinate can be obtained:
22 2 11∂∂ψ ∂ ψψ ∂ ∇=ψ r + + =0 (12) rr∂∂ r r22 ∂θ ∂ z 2 By separation of variables, the Laplace equation (12) can be decomposed into three differential equations with decoupled components, and different solutions are obtained corresponding to different values of parameters (see Appendix). Finally, the general solutions of the Laplace equation are in the non-singular form ψθin (,rz , )(finite at r = 0 for interior problems) and singular form ψθex (,rz , ) (infinite at r = 0 for exterior problems):
∞ ermmcos( mθθ )+ er sin( m ) ψθ(,rza , )=++ az 12mm in 01 02 ∑ mm m=1 ++e34mm zrcos( mθθ ) e zr sin( m ) ∞ c1nmcosh( nz ) J 02 ( nr )+ c sinh( nz ) J 0 ( nr ) + ∑ n=1 ++c3mmcos( nz ) I 04 ( nr ) c sin( nz ) I 0 ( nr )
d12mn cos( mθ )cosh( nz ) Jm ( nr ) + d mn sin(mθ )cosh( nz ) Jm ( nr ) ∞∞ ++d34mn cos( mθθ )sinh( nz ) Jm ( nr ) dmn sin( m )sinh( nz ) Jm ( nr ) + ∑∑ ++θθ nm=11 = d56mn cos( m )cos( nz ) Im ( nr ) dmn sin( m )cos( nz ) Im ( nr ) ++d78mn cos( mθθ )sin( nz ) Im ( nr ) dmn sin( m )sin( nz ) Im ( nr ) (13) and
−− ∞ ermmcos( mθθ )+ er sin( m ) ψθ(rza , , )=++ ln razr ln 56mm ex 03 04 ∑ −−mm m=1 ++e78mm zrcos( mθθ ) e zr sin( m ) ∞ c5nmcosh( nz ) Y 06 ( nr )+ c sinh( nz ) Y 0 ( nr ) + ∑ n=1 ++c7mmcos( nz ) K 08 ( nr ) c sin( nz ) K 0 ( nr )
d9mn cos( mθ )cosh( nz ) Ym (nr )+ d10mn sin( mθ )cosh( nz ) Ym ( nr ) ∞∞ ++d11mn cos( mθθ )sinh( nz ) Ym ( nr ) d12mn sin( m )sinh( nz ) Ym ( nr ) + ∑∑ ++θθ nm=11 = d13mn cos( m )cos( nz ) Km ( nr ) d14mn sin( m )cos( nz ) Km ( nr ) ++d15mn cos( mθθ )sin( nz ) Km ( nr ) d16mn sin( m )sin( nz ) Km ( nr ) (14) Substituting the non-singular solution, Eq. (13), into Eq. (7) generates the displacement expression in interior domain: uin =4(1 −v ) Bin + RB ⋅∇in − RB ∇⋅ in (15) while the displacement field in the exterior domain is obtained through substituting Eq. (14) and Eq. (13) into Eq. (7) and Eq. (6), respectively:
12 uuuex= ex + ex 1 111 uex =4(1 −v ) Bex + RB ⋅∇ex − RB ∇⋅ ex (16) 2 22 uex =4(1 −v ) Bex −∇ RB ⋅ ex Then strain-displacement and stress-strain relations Eqs. (2-3) are employed to calculate the stress components. The derivation process is very complicated and achieved using Wolfram Mathematica, which is not elaborated here. In order to determine the unknown coefficients in Eqs. (13-14), the collocation method is adopted to match the tractions ( tn= ⋅σ ) or displacements ( u ) to prescribed boundary values, instead of using the term-to-term matching process used by [19-20]. Thus, a system of linear algebraic equations are established as: Ay= b (17) in which y is a vector of unknown coefficients. Mostly a larger number of equations are established compared to the number of the unknown coefficients, so that Eq. (17) can be solved in a least square sense. 3. Scaling techniques for the ill-conditioned system of equations
As shown in Eqs. (13-14), both the interior and exterior parts contain various kinds of basis functions such as polynomial, trigonometric, Bessel, and exponential functions. These basis functions lead to severely ill-conditioned system of equations, because both very large and very small valued terms exist in the coefficient matrix A of Eq. (17). Thus, necessary scaling techniques need to be used to reduce the condition number of A . Step 1:
Firstly, a characteristic length r0 [26] is introduced:
η = rr0 , ξ = zr0 (18) by which the Laplace equation is revised as:
11∂∂ψ ∂22 ψψ ∂ η + +=0 (19) η∂∂x η ηθ22 ∂ ∂ 2 ξ It can be easily seen that the revised Laplace equation has the same form as Eq. (12).
The values of r0 are determined by whether the non-singular solutions (13) or singular solutions (14) are considered:
a) r0 = the largest distance from the local boundary of domain to the source point for uin
1 of Eq. (13) and uex of Eq. (14);
2 b) r0 = the smallest distance from the local boundary of domain to the source point for uex of Eq. (14). It should be noted that the source point is located at the center of domain. In this way, both the absolute values of η and ξ are confined between 0 and 1, and the condition number of A can be reduced. In order to keep consistency, we still keep the original
notations with r→ rr0 and z→ zr0 . Step 2: Even though the first characteristic length is introduced to the scale the harmonic functions, it is found out by numerical examples that Eq. (17) can still be a highly ill- conditioned system of equations, especially when the number of terms in basis functions increases. Thus, a recently proposed multiple scaling technique [10-11] is adopted to reduce the condition number and to stabilize the solutions of Eq. (17).
A multiple-scale characteristic length Lj are adopted to scale each column of the ill- conditioned matrix A , as is shown in [11]:
2 LAj = ∑ ij (20) i by which Eq. (17) can be rewritten as a new system of equations: Cx= b (21) Eq. (21) is related to Eq. (17) by
AA11 12 AA13 1×nn LLL123 Lnn AAAA yL× 21 22 23 2×nn 11 LLL L × 123 nn yL22 C = AAA A× , x = yL× (22) ij 31 32 33 3 nn j 33 LLL123 Lnn yL× nn nn AAAmm××123 mm mm × Amm × nn LLL123 Lnn
and C , x and b have the same dimensions with that of Eq. (17). In this way, each column of the coefficient matrix has the same norm, and the ill-conditioned system equations can be regularized. After applying the two scaling techniques, Eq. (22) can be solved in the sense of least square,
−1 x= ( CCTT) Cb (23)
where the unknown coefficients are obtained, leading to the solutions of the displacement, strain, and stress fields.
4. Numerical examples
Several examples are given in the present work to present the proposed 3D Trefftz collocation method. The completeness of the trial functions, the effect of the scaling techniques, and the accuracy of solutions are also discussed in this section.
Figure 1 A hollow cylinder with just matrix (m) phase or a composite cylinder with fiber (f) and matrix (m) phases.
4.1 A hollow cylinder
In the first case, we use the proposed method to study the stress distribution of a long hollow cylinder under uniform external and interior pressures, see Fig. 1. The material properties considered here are Ev()mm=1, () = 0.35 . The following geometric parameters are chosen to simulate a unidirectionally long cylinder: ab=1,4,20 = h = . Thus, the domain of interest is defined as: Ω∈≤≤≤≤−≤≤{arb , 0θπ 2 , h 2 zh 2} (24) while the boundaries of the domains in this case can be described as
Γ∈=≤≤−≤≤1 {ra , 0θπ 2 , h 2 zh 2}
Γ∈=≤≤−≤≤2 {rb , 0θπ 2 , h 2 zh 2} (25)
Γ∈3 {arb ≤ ≤ , 0 ≤θπ ≤ 2 , z =± h 2} Under the generalized plane strain assumption, the displacement field for such a hollow cylinder can be expressed as:
1+ν ()m ab 22 pa22−− pb p p ur(,)θν=−+[(1 2()m ) a br ab] r E() b2− a 2 ab22 r (26)
urθ (,θ )= 0 and the traction components are pa22−− pbab22 p p σθ(,r )= a b− ab rr ba22−− ba 22 r 2 pa22−− pbab22 p p σθ(,r )= a b+ ab (27) θθ ba22−− ba 22 r 2 pa22− pb σθν(,r )= 2 ()m ab zz ba22− while others are zeros. The explicit analytical displacement expressions are used as the
displacement boundary condition applied at Γ1 and Γ2 , and stress solutions are employed
as the traction boundary conditions applied at Γ3 , which are matched at collocation points with the proposed displacement and traction components. The number of collocation points should be equal or larger than the unknown coefficients. In this example, mn=2, = 2 are used for the harmonic terms, leading to 300 unknown coefficients need to be solved. Thus, 800 collocation points are uniformly distributed on the inner and outer side surfaces of the cylinder, and 400 collocation points are distributed on the top and bottom surfaces in a fan-shape manner, Fig. 2. The system of linear equations for the unknowns is similar to what is shown in Eq. (21), and the condition numbers are generated and compared in Table 1 among different steps of scaling procedure.
Figure 2 Collocation points distributed at surfaces of a hollow cylinder.
Before step 1 After step 1 After step 2 7.784e10 9.415e9 8.098e3 Table 1 Condition numbers generated before and after scaling steps for the problem of the hollow cylinder.
It can be easily seen that the condition number is only slightly reduced after step 1, while it is significantly reduced after step 2, which means that the multiple scaling technique works effectively for the 3D Trefftz collocation method. In order to study the accuracy of the Trefftz solution, the analytical solutions for stress fields (Eq. 27) are used
for comparison. Firstly we just plot σ xx (xyz , ,= 0) given by analytical expressions and computed by the TCM (Fig. 3), in which no visible difference can be seen within the same scale. The errors of all six stress components in the domain are then projected onto the x-y plane in Fig. 4, defined as
22 e=1 −=∑∑σσCij Aij ij, 1, 2, 3 (28) where the subscripts “C” and “A” stand for the collocation solution and analytical results, respectively. The errors generated between both methods are extremely small. It should also be noted that the matrix without scaling has to be invert using “pinv” command provided by Matlab program, otherwise “ill conditioning” warning signal appears and not enough accurate results are generated. However, only back slash “\” is needed in the new system of equations after the scaling steps. The completeness of the solution contributes to the accuracy of the stress distributions obtained through collocation technique. Here we also generate the stress distributions and relative errors by just employing the non-singular terms as the solution of the hollow cylinder with same dimensions and under same loading conditions. Fig. 5 illustrates the stress component σ xx (xyz , ,= 0) obtained in this case and the relative errors against
analytical solutions, from which it is easily noted that σ xx has similar pattern as Fig. 3 but in a totally different scale, and the corresponding errors are significant. Therefore, the incomplete Trefftz solution cannot represent the 3D elasticity problems within the exterior domains.
(a) Analytical solution
(b) TCM
Figure 3 Stress distributions σ xx (xyz , ,= 0) by (a) Analytical expressions and (b) TCM for the hollow cylinder problem.
Figure 4 Surface plot of errors between Trefftz collocation results and analytical solutions for the hollow cylinder problem.
(a)
(b)
Figure 5 Stress distributions σ xx (xyz , ,= 0) by employing only non-singular terms (a) and the errors of stress components against analytical solutions (b).
4.2 A composite cylinder
Composite cylinders subjected to outer radial pressure pb are then investigated. Fig. 1 illustrates the composite cylinder with two phases: the fiber and the matrix. Geometric parameters ab=1,4,20 = h = are still considered. Material properties of the fiber phase
are E ()f =10 , v()f = 0.2 , while the properties of matrix are still Ev()mm=1, () = 0.35 . The object domain of the composite cylinders is divided into two part: Ω∈≤≤≤≤−≤≤{0ra , 0θπ 2 , h 2 zh 2} f (29) Ω∈≤≤≤≤−≤≤m {arb , 0θπ 2 , h 2 zh 2} The boundaries of the domains are then presented as
Γ∈=≤≤−≤≤1 {rb , 0θπ 2 , h 2 zh 2}
Γ∈2 {arb ≤ ≤ , 0 ≤θπ ≤ 2 , z =± h 2} (30)
Γ∈3 {0 ≤ra ≤ , 0 ≤θπ ≤ 2 , z =± h 2} and the continuity conditions between fiber and matrix are confined within
Γ∈=≤≤−≤≤4 {ra , 0θπ 2 , h 2 zh 2} (31) The analytical solution can be obtained from the system of equations
2 1−− 110aAf ()f () f () f () m () m () mm () () m 2 E(1+−νν )(1 2 ) −E (1 +− νν )(1 2 )E (1 + ν ) aAm =0 ()mm+−νν () () m − () mm + ν () 2 − 0 E(1 )(1 2 )E (1 ) bBmb P (32)
where Af is the coefficient for the fiber, and Am , Bm are the coefficients for the matrix, which need to be solved before the analytical solutions are obtained. It should be noted that the displacement boundary conditions are applied at the side surface of the cylinder Γ1 : u() r= b = Ab + B b r mm (33) urθ (= b )0 = and the traction boundary conditions are applied at the top and bottom surface of the cylinder
()m σθνzz (,rA )= 2 m at Γ2
()f σθνzz (,rA )= 2 f at Γ3 (34) The conditions of displacement continuity and traction reciprocity at the fiber-matrix interface are also enforced by the collocation method at Γ4 :
mf mf uraii()()= = ura = and traii()()= = tra = (35) Establishing Eqs. (33-35) yields the system of equations for both fiber and matrix unknown coefficients, where Eq. (17) can be re-expressed as
A[Ffm ,F ]= b (36)
and A is the matrix where elements are functions of structural geometries and material
properties. Ff and Fm are the vectors with unknown coefficients of fiber and matrix,
respectively. By letting mn=2, = 2 , 300 unknown coefficients for the matrix and 150
unknown coefficients for the fiber are created. b is the vector for boundary conditions. In
this situation, 400 points are distributed at Γ1 and Γ2 , 200 points are distributed at Γ3 ,
and another 400 points are used for the continuity conditions at Γ4 , as shown in Fig. 6.
Figure 6 Collocation points distributed at surfaces of composite cylinders.
Before step #1 Before step #2 After scaling 1.382e14 7.791e13 6.206e5 Table 2 Condition numbers generated without or with scaling procedure for composite cylinders.
Table 2 lists the condition numbers generated using the scaling process. Similar with the situation of hollow cylinder, the condition number greatly decreases by taking the scaling procedure, making the calculated results more accurate and stable. The stress distributions generated by using analytical result and TCM are compared in Fig. 7, where good agreement is still fulfilled, and minimum relative errors are observed in Fig. 8. A
closer examination of the interfacial stresses σ rr , σ rθ , σθθ are demonstrated at the perimeter of the fiber ( ra= ) in the mid plane of z-axis ( z = 0 ) in Fig. 9. The stress components are generated from both fiber and matrix expressions. As suggested by
continuity conditions of Eq. 35, σ rr , σ rθ generated from TCM are totally matched from
both sides of the perimeter, while σθθ shows different trends. In addition, the analytical solutions always predict same behavior with their corresponding TCM results.
(a) Analytical result
(b) TCM
Figure 7 Stress distributions σ xx (xyz , ,= 0) generated by (a) Analytical expressions; (b)
TCM technique for hollow cylinder under radial pressure Pb = −2 .
Figure 8 Surface plot of errors between analytical and TCM with scaling procedures for composite cylinders under radial pressure Pb = −2 .
Figure 9 Interfacial Stress distributions σ rr (r= az , = 0) , σ rθ (r= az , = 0) , σθθ (r= az , = 0) generated by TCM and analytical solutions at the perimeter of the fiber.
4.3 A solid sphere
The third example is a solid sphere under outer pressure pb = −2 . The radius of the sphere
is b = 4 , while the material properties are Ev=1, = 0.35 . The domain of interest is defined as Ω={rb ≤ ,0 ≤θ ≤ 2 π ,0 ≤ φπ ≤ } (36) and the boundary of the domain is defined as Γ={rb = ,0 ≤θ ≤ 2 π ,0 ≤ φπ ≤ } (37) The analytical solutions of the displacement and stress components are: u =−−(1 2ν ) E ⋅ Pr rb (38) uθ = 0 and
σ rr= −P b
σθθ = 0 (39)
σφφ = 0
respectively. Eqs. 38-39 can be used to apply boundary conditions at Γ . In the present case, only the non-singular terms are employed to match the boundary conditions, which are used to deal with the interior problem. 400 collocation points are used to match the boundary conditions for the functions with 150 unknown coefficients with mn=2, = 2 , Fig. 10.
Figure 10 Collocation points distributed at surfaces of the solid sphere.
Again, the condition number is largely decreased using the scaling technique (Table 3), even though the equation system is not severely badly scaled before the scaling process due to the defined dimensions of the sphere.
Before scaling After scaling 1.318e8 7.110e5 Table 3 Condition numbers generated without or with scaling procedure for solid sphere.
The errors of all six stress components, projected on x-y plane, are illustrated as a surf plot in Fig. 11, revealing the high accuracy of the TCM solutions even in the spherical domain.
Figure 11 Surface plot of errors between analytical and TCM results with scaling procedures for solid sphere under radial pressure Pb = −2 .
The convergence of collocation technique is also studied in the present scenario. Different numbers of collocation points (more than the number of coefficients) are tested in the range of [160, 600] with the increment of 20. It is learnt (not shown) that the convergence of the collocation technique is fairly rapid even though minimum number of required points is used. 4.4 Bending of a circular bar
Next scenario is another classical elasticity problem dealing with the bending of a circular solid bar with the raidus of a = 4 and height of h = 20 . The bar is fixed at one end of zh= − 2 and applied by a force P =1 at the other end of zh= 2 . The loading is parallel to the x axis. Thus, the domain is defined as Ω∈≤≤≤≤−≤≤{0ra , 0θπ 2 , h 2 zh 2} (40) and the boundary conditions are divided into side surface and end surfaces of the bar:
Γ∈=≤≤−≤≤1 {ra , 0θπ 2 , h 2 zh 2}
Γ∈2 {0 ≤ra ≤ , 0 ≤θπ ≤ 2 , z =± h 2} (41) The similar problems have been studied by several authors, and Timoshenko and Goodier [27] provided the explicit expressions for stresses expressed in Cartesian coordinate: (3+− 2νν )P 1 2 σ (,xy )= (a22 −− x y 2) xz 8(1++νν ) I 3 2 (42) (1+ 2ν ) Pxy σ (,xy )= − yz 4(1+ν ) I
and other components are zeros. The traction boundary conditions are applied at Γ1 and
Γ2 while the displacement boundary conditions are applied at the bottom side of the
surface Γ2 in order to fix the bar from rigid body movement. 400 collocation points are
distributed along Γ1 and Γ2 , respectively, Fig. 12. It is found that 150 harmonic terms ( mn=2, = 2 ) are enough to represent the stress distributions: Here we show the component σ xz (xyz , ,= 0) , Fig. 13. From the analytical expressions in Eq. 42, one can easily derives that at the center of the bar ( xy=0, = 0 ), the maximum stress is obtained as:
(3+ 2ν )Pa2 σ (xy= 0, = 0) = (43) xz max 8(1+ν ) I which can be calculated and compared with TCM result in Table 4. A good agreement is finally generated with the error at the fourth decimal place.
Figure 12 Collocation points distributed at surfaces of the solid cylinder.
(a) Analytical result
(b) TCM
Figure 13 Stress distributions σ xz (xyz , ,= 0) generated by (a) Analytical expression; (b) TCM result for bending problem of a circular bar.
Analytical solution TCM result 0.2181 0.2178
Table 4 Maximum shear stress σ xz generated by analytical solution and TCM result for the bending problem of a circular bar.
4.5 Torsion of a circular bar
Finally, the torsion problem of a circular bar with same dimension as Section 4.4 is investigated. The domain of interests and boundary locations are totally same as before. It
is assumed that the bar is twisted by couples at the ends, with the magnitude of M t =1000 as the resultants of the forces. Timoshenko and Goodier [27] still derived the analytical expressions of shear stress components: 2My σ (,xy )= − t xz π 4 a (44) 2Mx σ (,xy )= t yz π a4
which are still used to apply at the boundaries Γ1 and Γ2 in Eq. 41. Same collocation points are used to establish the system of equations for the unknown coefficients. The existing stress distributions are still compared with the analytical solutions in Eq. 44, which generates well matched comparison with a maximum relative error of 2e-13. From Eq. 44, it is evident that he maximum stress is at the points intersected by x or y axis and
3 the boundaries, with a absolute value of 2Mat π . Table 5 compares the values between analytical solution and the TCM result and a good agreement is obtained, showing the accuracy of the Trefftz method and collocation technique.
Analytical solution TCM result 9.947 9.947
Table 5 Maximum shear stress σ xz generated by analytical solution and TCM result for the torsion problem of a circular bar.
5. Conclusions
A Trefftz collocation method is developed to solve 3D linear elasticity problems. The trial functions are expressed in terms of the Papkovich-Neuber solution, with cylindrical harmonics as potential functions. The Trefftz trial functions, which contain both non- singular and singular terms, can deal with interior problem as well as exterior problems. In order to relieve the problem of ill-conditioning, a characteristic-length is firstly introduced into the harmonic terms, and another multiple scaling technique is then adopted to effectively further reduce the condition numbers. Finally, several numerical examples are solved to test the validity of the proposed method, where collocation method is used to enforce boundary conditions. It is demonstrated: (1) singular terms in the Trefftz trial function are necessary for exterior problems; (2) the multiple-scaling technique can effectively relieve the problem of ill-conditioning; (3) the proposed Trefftz method can solve 3D linear elasticity problems with high accuracy and efficiency. The proposed TCM will facilitate our future work of implementing Trefftz trial functions in the framework of computational grains [16-17] for high-performance simulation and design of fiber-reinforced composite materials. Appendix
Applying separation of variables using ψθ(,r , z )= f () rg ()() θ hz, substituting it into (8)
and multiplying by r2 f() rg ()()θ hz yields
1∂∂f 11 ∂22 gh ∂ r + +=0 (A1) rf∂∂ r r r22 g ∂θ h ∂ z 2 The z-dependent term can be immediately isolated as 1∂∂f 11 ∂22 gh ∂ r +=−=λ (A2) rf∂∂ r r r22 g ∂θ h ∂ z 2
1 ∂2h For − =λ , the characteristic polynomial is r 2 +=λ 0 which leads to r =±−λ , hz∂ 2 and
2 a) If λ < 0 , let λ = −n : h( z )= Enn cosh( nz )+ F sinh( nz ) ;
b) If λ = 0 : hz()= E00 + Fz;
2 c) If λ > 0 : let λ = n and r= ± in: h( z )= E−−nn cos( nz ) + F sin( nz ) . Eq. (A2) can be then manipulated as
2 rf∂∂2 1 ∂ g rr−=−λκ= (A3) fr∂∂ r g ∂θ 2
1 ∂2 g For − =κ , the characteristic polynomial is r 2 +=κ 0 , and r =±−κ : g ∂θ 2
2 a) If κ < 0 , let κ = −m : g(θ )= Cmm cosh( mD θθ )+ sinh( m ) ;
b) If κ = 0 : g()θθ= CD00 + ;
2 c) If κ > 0 : let κ = m and κ = ±im : g(θ )= C−−mm cos( mF θθ )+ sin( m ) .
However, the solution is required as periodic to ensure g()θ is single valued:
gg()θ= ( θπ + 2)
Thus, the solution for g()θ can be either
g(θ )= C−−mm cos( mF θθ )+ sin( m ) or gC()θ = 0 while m is an integer. What is left is the radial solution in r direction: ∂∂2 ff1 κ + +−(λ − )0f = (A4) ∂∂r22 rr r whose solutions depend on the values of both λ and κ :
a) λ = 0 and κ = 0 , then fr( )= A00 + B 00 ln r;
2 mm− b) λ = 0 and κ > 0 (κ = m ), then fr()= Ar00mm + B r
c) λ < 0 ( λ = −n2 ) and κ > 0 (κ = m2 ), then ∂∂22ff1 m + +−(nf2 )0 = ∂∂r22 rr r which is Bessel differential equation, and leads to Bessel functions of first and second kinds:
f() r= A−−mn J m () nr + Bmn Y m () nr
d) λ < 0 and κ = 0 , then f() r= A−−nn0 J 0 () nr + B00 Y () nr
e) λ > 0 ( λ = n2 ) and κ > 0 (κ = m2 ), then ∂∂22ff1 m + +−(nf2 − )0 = ∂∂r22 rr r which is the modified Bessel differential equation, the solutions of which are modified Bessel functions of first and second kinds:
f() r= Amn I m () nr + Bmn K m () nr
f) λ > 0 and κ = 0 , then f() r= Ann00 I () nr + B0 K 0 () nr Based on the analyses of the three separated differential equations (A2-A4), the solutions of the Laplace equation Eq. () usually falls in the following categories:
1) λκ=0, = 0 : ψθ1(r , , z )=+ ( E0 FzA 0 )( 00 + B 00 ln r ) =++ a01 az 02 a 03 ln r + az04 ln r (A5)
2) λ = 0 and κ > 0 (κ = m2 ):
mm− ψθ2(r , , z )= ( E00++ Fz )[ C−−m cos( mθ ) F m sin( m θ )]( A0 mm r + B 0 r ) mm m m =++ermermezrmezrm1mcos(θθ ) 23 mm sin( ) cos( θ ) + 4 m sin( θ ) (A6) −−mm − m − m +++ermermezrmezrm5mcos(θθ ) 67 mm sin( ) cos( θ ) + 8 m sin( θ ) 3) λ ≠ 0 and κ = 0 , then
ψθ3(r , , z )=++ [ En cosh( nz ) F n sinh( nz )] C0 [ A nn 0 J 0 ( nr ) B00 Y ( nr )] ++[E cos( nz ) F sin( nz ) ] C [ A I ( nr )+ B K ( nr )] −−n n 0 mn m mn m (A7) =c1ncosh( nz ) J 02 ( nr ) + cm sinh( nz ) J 03 ( nr ) ++ cm cos( nz ) I mmm ( nr ) c4 sin( nz ) I ( nr ) c5nmcosh( nz ) Y 06 ( nr )+ c sinh(nz)() Y07 nr++ cm cos() nz K mm () nr c8 sin() nz K m () nr 4) λ ≠ 0 and κ > 0 , the
ψθ4 (,rz , )=
[En cosh( nz )+ Fn sinh( nz )][ C−−m cos( mθθ ) ++ Fm sin( m )][ Anm J m ( nr ) Bnm Y m ( nr )]
++[E−−−n cos( nz ) Fn sin( nz ) ][ Cm cos( mθθ )+ F −m sin( m )][ Amn I m ( nr ) + Bmn K m ( nr )]
= d12mn cos( mθθ )cosh( nz ) Jm ( nr )+ dmn sin( m )cosh( nz ) Jm ( nr )
+d34mn cos(mθθ )sinh( nz ) Jm ( nr )+ dmn sin( m )sinh( nz ) Jm ( nr )
++d56mn cos( mθθ )cos( nz ) Im ( nr ) dmn sin( m )cos( nz ) Im ( nr )
++d78mn cos( mθθ )sin( nz ) Im ( nr ) dmn sin( m )sin( nz ) Im ( nr )
++d9mn cos( mθθ )cosh( nz ) Ym ( nr ) d10mn sin( m )cosh( nz ) Ym (nr)
++d11mn cos( mθθ )sinh( nz ) Ym ( nr ) d12mn sin( m )sinh( nz ) Ym ( nr )
++d13mn cos( mθθ )cos( nz ) Km ( nr ) d14mn sin( m )cos( nz ) Km ( nr )
++d15mn cos( mθθ )sin( nz ) Km ( nr ) d16mn sin( m )sin( nz ) Km ( nr )
(A8)
ψ1 ,ψ 2 , ψ 3 ,ψ 4 are then reorganized as nonsingular and singular solutions shown in the main text.
References
[1] Trefftz E. Ein Gegenstück zum Ritzschen Verfahren. Proc 2nd Int Cong. Appl Mech, Zurich 1926, 131–137. [2] Kita E, Kamiya N. Trefftz method: an overview. Adv Eng Softw 1995; 24: 3-12. [3] Li Z-C, Lu T-T, Hu H-Y, Cheng AH-D. Trefftz and collocation methods. WIT Press, Southampton/Boston 2008. [4] Dong L, Atluri SN. A simple multi-source-point Trefftz method for solving direct/inverse SHM problems of plane elasticity in arbitrary multiply-connected domains. CMES: Comput Model Eng Sci 2012; 85(1): 1-43. [5] Cheung YK, Jin WG, Zienkiewicz OC. Solution of Helmholtz equation by Trefftz method. Int J Numer meth Eng 1991; 32(1): 63-78. [6] Chang J, Liu R, Kuo S, Yeih W. Application of symmetric indirect Trefftz method to free vibration problems in 2D. Int J Numer Meth Eng 2003; 56: 1175-1192. [7] Li Z-C, Lu T-T, Tsai H-S, Cheng AHD. The Trefftz method for solving eigenvalue problems. Eng Anal Bound Elem 2006; 30: 292-308. [8] Kita E, Ikeda Y, Kamiya N. Indirect Trefftz method for boundary value problem of Poisson equation. Eng Anal Bound Elem 2003; 27: 825-833. [9] Liu CS, Yeih W, Atluri SN. On solving the ill-conditioned system Ax=b: general- purpose conditioners obtained from the boundary-collocation solution of the Laplace equation, using Trefftz expansions with multiple length scales. CMES: Comput Model Eng Sci 2009; 44(3): 281-312. [10] Ku C-Y. On solving three-dimensional Laplacian problems in a multiply connected domain using the multiple scale Trefftz method. CMES Comput Model Eng Sci 2014; 98(5): 509-541. [11] Ku C-Y, Kuo C-L, Fan C-M, Liu C-S, Guan P-C. Numerical solution of three dimensional Laplacian problems using the multiple scale Trefftz method. Eng Anal Bound Elem 2015; 50: 157-168. [12] Kita E, Kamiya N, Iio T. Application of a direct Trefftz method with domain decomposition to 2D potential problems. Eng Anal Bound Elem 1999; 23: 539-548. [13] Jin WG, Cheung YK, Zienkiewicz. Trefftz method for Kirchhoff plate bending problems. Int J Numer Meth Eng 1993; 36(5): 765-781. [14] Jirousek J, Leon N. A powerful finite element for plate bending. Comput Methods Appl Mech Eng 1977; 12: 77-96. [15] Jirousek J, Teodorescu P. Large finite elements method for the solution of problems in the theory of elasticity. Comput Struct 1982; 15(6): 575-587. [16] Dong L, Atluri SN. Development of 3D T-Trefftz voronoi cell finite elements with/without spherical voids &/or elastic/rigid inclusions for micromechanical modeling of heterogeneous materials. CMC: Comput Mater Con 2012; 29 (2): 169-211. [17] Dong L, Atluri SN. Development of 3D Trefftz voronoi cells with ellipsoidal voids &/or elastic/rigid inclusions for micromechanical modeling of heterogeneous materials. CMC: Comput Mater Con 2012; 30 (1): 39-81. [18] Qin Q-H. Trefftz finite element method and its applications. Appl Mech Rev 2005; 58: 316-337. [19] Chau KT, Wei XX. Finite solid circular cylinders subjected to arbitrary surface load. Part I – Analytic solution. Int J Solids Struct 2000; 37: 5707-5732. [20] Serati M, Alehossein H, Williams DJ. 3D elastic solutions for laterally loaded discs: Generalised Brazilian and point load tests. Rock Mech Rock Eng 2014; 47: 1087-1101. [21] Liu CS. A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length. CMES: Comput Model Eng Sci 2007; 21(1): 53-65. [22] Liu CS. An effectively modified direct Trefftz method for 2D potential problems considering the domain’s characteristic length. Eng Anal Bound Elem 2007; 31(12): 983- 993. [23] Lurie AI. Theory of Elasticity. Springer 2005. [24] Papkovish PF. Solution Générale des équations differentielles fondamentales d'élasticité exprimée par trois fonctions harmoniques. Compt Rend Acad Sci Paris 1932; 195: 513–515.
[25] Neuber H. Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. J Appl Math Mech - USSR 1934; 14(4): 203-212. [26] Steven GP. The eigenvalue problem for hollow circular cylinders. Int J Eng Sci 1973; 11: 795-810. [27] Timoshenko SP, Goodier JN. Theory of elasticity. McGraw-Hill 1970.