Chapter 7
Unresolved problems of adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics Grzegorz Zboi´nski
Abstract In this chapter of the book we present some chosen problems of adaptive hierarchical modelling and adaptive hp-analysis of problems of computational solid mechanics. We consider simple and complex structures, i.e. structures described by one mechanical model or more than one mechanical model, respectively. We are interested in three fundamental problems of solid mechanics: the equilibrium (static) problem, eigenvalue (free vibration) problem, and stationary forced vibration problem as well. In the context of static analysis, we consider problems faced while applying: the hierarchical models, hp-approximations, residual error estimation methods, and three- or four-step non- or iterative adaptive strategies, oriented on the target admissible value of the modelling and approximation errors. We also address possibilities and difficulties of generalization of the mentioned techniques onto free and forced vibration analyses. In this contribution we are interested mainly in still unresolved or open problems. We will show some ways, either potentially available or checked by us numerically, to cope with all the mentioned issues.
1 Introduction In this book chapter we would like to discuss some theoretical and implementation difficulties of the adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics. We present these difficulties in the context of our uniform methodology that can be applied to the equilibrium, free vibration and stationary forced vibration problems. This methodology [19] can be applied to both, simple or complex structures, i.e. structures of simple or complex mechanical deGrzegorz Zboi´nski Polish Academy of Sciences, Institute of Fluid Flow Machinery, ul. Fiszera 14, 80-952 Gda´nsk, Poland, e-mail:
[email protected] University of Warmia and Mazury, Faculty of Technical Sciences, ul. Oczapowskiego 11, 10-736 Olsztyn, Poland This preprint was published as: G. Zboi´nski. Unresolved problems of adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics. In: Kuczma M., Wilma´nski K. (eds.) Computer Methods in Mechanics - Lectures of the CMM 2009. Series: Advanced Structural Materials, vol. 1, chapter 7, pp. 113-147. Springer Verlag, Berlin 2009.
1
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Grzegorz Zboi´nski
scription, based on one or more mechanical models, respectively. These two descriptions can be applied within both, simple and complex geometries. Our simple geometries, as well as different parts of complex geometries, can be of either solid, shell, or transition character. Note that our approach is much more general than the approaches of the predecessors, who usually consider one problem of solid mechanics, single mechanical model, and one type of the applied geometry. In other words, we search for one generalizing approach, suitable for a wide range of problems, rather than for specialized approaches, assigned for the specific mechanical problems.
1.1 Considered problems of solid mechanics As mentioned above, we consider three fundamental problems of solid mechanics: the static equilibrium problem, and the dynamic problems of free and stationary forced vibrations.
1.1.1 Static equilibrium problem Let us start with the equilibrium problem. In this case we search for the static solution in displacements u = u(x). The local formulation within the elastic body V under consideration (Fig. 1) is
σi j, j + f0i (x) = 0, εi j = 1/2(ui, j + u j,i ), σi j = Di jkl εkl , x ∈ V,
(1)
where Di jkl , i, j, k, l = 1, 2, 3 is the elastic constants tensor, σi j and εkl are stress and strain tensors, f0i is the vector of static body forces, while ui is the ith component of the vector u. The above set has to be completed with the traction and kinematic boundary conditions on parts SP and SD respectively, of the body surface S
σi j n j = p0i , x ∈ SP , ui = d0i , x ∈ SD ,
(2)
with n j denoting the vector normal to this surface, p0i standing for the static surface tractions, and d0i being the components of the prescribed surface displacements. In order to obtain the finite element equations of the problem we have to take advantage of the variational formulation corresponding to the above local formulation. The former formulation reads Z
V
Di jkl vi, j uk,l dV =
Z
V
vi f0i dV +
Z
SP
vi p0i dS,
(3)
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
3
f
0
SD SP
V
M M
Fig. 1 The elastic body under consideration (the static case)
p
0
with vi standing for the admissible values of the trial displacement functions, conforming to the kinematic boundary conditions. The vector form of the finite element formulation, obtained from (3) after the discretization and the introduction of the hpq-interpolation shape functions, are Kqhpq = F0 ,
(4)
where K is the global nodal stiffness matrix, F0 is the global nodal static forces vector, while qhpq stands for the global nodal vector of the displacement dofs correM sponding to the hpq finite element formulation.
1.1.2 Free vibration problem In the case of the free vibration problem (eigenproblem) of the elastic body, the solution, we search for, takes the form u = a(x)e jω0 t , where a(x) represents free vibration amplitudes, ω0 is the natural frequency of the body, t is the time variable, while j is the imaginary unit. The local formulation reads now ..
σi j, j = ρ ui , εi j = 1/2(ui, j + u j,i ), σi j = Di jkl εkl , x ∈ V, ..
(5) jω t
with ρ standing for the material density and ui = −ω02 ai 0 being the second time derivatives of the displacements (the acceleration components). The usual boundary conditions for the free vibration problem are
σi j n j = 0, x ∈ SP , ui = 0, x ∈ SD ,
(6)
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Grzegorz Zboi´nski
The corresponding variational formulation, necessary for derivation of the finite element equations, and expressed through the amplitudes, is Z
V
Di jkl vi, j ak,l dV − ω02
Z
V
ρ vi ai dV = 0
(7)
In the above relation, the values vi represent the kinematically admissible values of the amplitudes, conforming to the boundary conditions (6) expressed through the amplitudes. Note, that now we search for the all eigenpairs (eigenvalues and eigenvectors), ω02 and a(x), fulfilling (7). In order to obtain the set of the finite element method relations, resulting from the above variational formulation, one has to perform division of the body into finite elements and introduce the hpq-interpolation. This leads to the discretized eigenproblem of N degrees of freedom. The characteristic equation, necessary for deter2 , n = 1, 2, ..., N, is mination of N eigenvalues ω0n det(K − ω02 M) = 0
(8)
where M stands for the global mass (inertia) matrix. The dynamic equilibrium equation, necessary for determination of the nth eigenvector of amplitudes is 2 M)qhpq = 0, (K − ω0n n T
qhpq Mqhpq n n =1
(9)
So as to obtain the unique values of the eigenvectors, the first equation (9) has been completed with the normalization condition (the second equation (9)), here corresponding to the M-orthonormal normalization. In the above relations the amplitude dofs vector qhpq has been employed. Note that the displacements corresponding to n nth mode of vibration can be calculated form the amplitudes and the natural frejω t quencies as qhpq n e 0n .
1.1.3 Forced vibrations We consider here the stationary forced vibration problem for which the superposition theorem is valid. In the case of a single force of vibration the local formulation reads .
..
σi j, j + α ρ ui + fi (x,t) = ρ ui , εi j = 1/2(ui, j + u j,i ), . σi j = Di jkl εkl + β ε i j , x ∈ V,
(10)
where the solution vector for displacements is u = a(x)e j(ω t+φ +ϕ ) , while the given body force is of the form f(x,t) = f0 (x)e j(ω t+φ ) , with f0 being the force amplitude vector, ω and φ standing for the given force frequency and force phase angle, and
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
5
ϕ denoting unknown phase angles of the vibration amplitudes. The second and first time derivatives of the displacement vector (the acceleration and velocity vectors) . .. are .ui = −ω ai e j(ω t+φ +ϕ ) , ui = jω ai e j(ω t+φ +ϕ ) , while the strain velocity is equal to ε i j = jω εi j . The coefficients α and β characterize the material dumping due to viscous internal friction and can be treated as equivalent to Rayleigh dumping coefficients. In the case of forced vibrations, the form of the surface tractions is p(x,t) = p0 (x)e j(ω t+φ ) , with p0 standing for the traction amplitude vector. The related traction boundary conditions, as well as the kinematic boundary conditions, are σi j n j = pi (x,t), x ∈ SP , ui = 0, x ∈ SD ,
(11)
The variational formulation for the stationary forced vibration problem, corresponding to (10) and (11), is Z
V
Di jkl vi, j ak,l dV + jω =
Z
V
Z
V
(β Di jkl vi, j ak,l + α ρ vi ai ) dV − ω 2
vi f0i e− jϕ dV +
Z
SP
Z
V
ρ vi ai dV
vi p0i e− jϕ dS,
(12)
with vi representing the kinematically admissible field of the displacement amplitudes. Note that the solution to (12) is searched in the complex functions domain. The physical unknowns of the problem are the amplitudes and their phase angles. After performance of the discretization and hpq-interpolation of the filed of unknowns, the matrix form of the finite element equations, corresponding to the above variational formulation, and assigned for any of m (m = 1, 2, ..., M) independent forces of vibration, is − j ϕm , (K + jωm C − ωm2 M)qhpq m = F0m e
(13)
where F0m is the global vector of the amplitudes of the mth force, while the global dumping coefficient matrix C is composed of two components, i.e. C = β K + α M. The total solution in displacements can be obtained with the superposition theorem, taking advantage of m consecutive solutions for nodal amplitudes qhpq m and phase j(ωm t+φm +ϕm ) . angles ϕm . The mth displacement contribution is qhpq e m
1.2 Considered types of complexity within the elastic bodies In order to retain the general character of the analysis, we allow geometrical and mechanical complexity of the elastic bodies, analyzed in three solid mechanics problems described in the previous subsections.
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Grzegorz Zboi´nski
solid geometry
transition geometry mid-surface shell geometry top surface mid-surface
bottom surface Fig. 2 Different types of the geometry
1.2.1 Geometrical complexity Let us start with exemplary simple geometries. Three of such geometries, the solid, shell, and transition ones, are shown in Fig. 2. The solid geometry is bounded with surfaces. The shell geometry is based on the mid-surface and thickness concepts. The transition geometry is partly defined with the equations of the bounding surfaces and partly described by means of the mid-surface equation and the thickness vector normal to this surface. The numerical representation of the simple shell geometry (the plate example) is shown in Fig. 4. Let us pass to complex geometry of the body. We deal with such geometry when more than one type of geometry is necessary for the description of the shape of the body. Such a situation is presented in Fig. 3, where the solid, shell, and transition geometries are employed within one body. The numerical representation of the complex geometry can also be seen in Fig. 5. A thorough mathematical description and the specific definitions of the above mentioned simple and complex geometries can be found in our work [13].
1.2.2 Mechanical complexity The structure of simple mechanical description is characterized with the application of one mechanical model. The complex mechanical description is a result of the necessity of application of more than one mechanical model within the structure. The application of one or more than one model refers to both, simple and complex geometries, of course. The typical situation for solid mechanics problems, where the solid, shell, and transition mechanical models are employed, is presented in Fig. 3. It
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
7
solid geometry
transition geometry
shell geometry 3D-model transition model Fig. 3 Complex mechanical description
shell model
can be seen from the figure that the division of the structure into zones (domains) of the different mechanical description is independent of the division into geometrical parts (members). It is obvious, however, that some models do not appear in certain parts, e.g. shell models are not suitable for solid parts of the body. The admissible neighbourhood of the different models and the appearance of the models in the different geometrical parts are explained in detail in [13]. The relations between the simple and complex geometries and the simple and complex mechanical descriptions are also illustrated in Fig. 4 and Fig. 5, where the complex mechanical models are applied within the simple and complex geometries, respectively. The first of these two figures shows the numerical representation of the symmetric quarter of the clamped plate, while in the second figure the numerical idealization of the symmetric quarter of the square floor supported by four columns is displayed. The models applied in both examples are: the theory of three-dimensional elasticity (3D) or the hierarchical shell models (MI) of the transverse order I (I = 1, 2, 3, ...), the first-order Reissner-Mindlin shell model (RM), and the solid-to-shell transition models, (3D/RM) or (MI/RM). The description of these models can be found in our work [13].
2 Assessed methodology In this section we would like to present the assessed methodology for the adaptive modelling and adaptive analysis of the simple and complex structures within solid mechanics. Our methodology covers the 3D-based hierarchical modelling and the hierarchical hp-, hpq-, and hpq/hp-approximations, which all allow generating the hierarchy of numerical finite element models of the local character. The methodology includes also the error estimation with the equilibrated residual method (ERM), and the error-controlled adaptivity, based on the three- or four-step adaptive strategies, with possible iterations within the h- and p-steps.
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Grzegorz Zboi´nski
model
3D MI
260 250 180
280
170 100 90 20
270 200 190
120
10
300
110 40
290 220 210
140
30
320 310 240
130 60
RM/3D RM/MI
230 160
50
150 80
RM
70
z x
y
Fig. 4 Numerical representation of the complex mechanical description within simple geometry
2.1 3D-based hierarchy of numerical models In this subsection we discuss three major components necessary for generation of our 3D-based hierarchy of numerical models assigned for the adaptive hierarchical modelling and hpq-adaptive analysis of the simple and complex structures within computational solid mechanics. The first component is the 3D-based approach which lies in the application of the same 3D degrees of freedom for any mechanical model. The second component is the hierarchical modelling, including such different mechanical models as: the 3D-elasticity, the higher-order hierarchical shell models, the first-order shell model, and the solid-to-shell or shell-to-shell transition models. The third component is the set of hpq hierarchical approximations applied to all models of the hierarchy of mechanical models. The combination of theses three components leads to the mentioned hierarchy of numerical models. Note that the hierarchical numerical models can be applied locally, i.e. on the finite element level. The idea is illustrated in Fig. 6, where in each element e of the vole
e
ume V , a different model M from the set of mechanical models M, the different size e e e h, and the different longitudinal and transverse approximation orders, p and q, can be chosen.
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
9
model
3D MI
160
150
140
180
170
130 100 90
RM/3D RM/MI
120 20 110 10
RM
z x
y
Fig. 5 Numerical representation of the complex mechanical description within complex geometry
SD SP
e
e
e e
M∈M, h, p, q e
V
Fig. 6 The idea of hierarchical numerical modelling
2.1.1 3D-based approach The idea of the 3D-based approach lies in application of only one type of degrees of freedom within all mechanical models included into the hierarchy of models. In our case only the three-dimensional degrees of freedom (three displacements at a point of the body) are employed, regardless of the model type. The conventional mid-surface displacements and rotations of the first-order shell model, as well as the generalized mid-surface dofs of the higher-order shell models are replaced with the equivalent through-thickness dofs. One more difference between both approaches is the direction of the dofs. In the conventional case we usually apply the local
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Grzegorz Zboi´nski
thickness
thickness
top
ξ' 3 1
d' 2j d' 0j
x'3
d'3n middle
x'2
d'2n
middle
u' j
d' 1j
d' 3j
top
ξ' 30 d' 0j ξ'31d' 1j
0
ξ'32 d' 2j
d'1n
bottom
u' j (ξ' 3 )
ξ'33 d' 3j x'1
-1
bottom − d' 1j
− d' 3j
d' 2j
d' 0j
Fig. 7 Mid-surface dofs (left) and the displacement field (right) of the conventional approach
directions, normal and tangent to the mid-surface, while in the case of the throughthickness dofs the global directions are employed. The equivalence of the local displacement fields u′j , j = 1, 2, 3 in the cases of the mid-surface and through-thickness dofs is shown in Fig. 7 and Fig. 8. This equivalence can also be expressed through the relation u′j =
I
∑ ξ3′
n=0
n ′n d j
I
=
∑ fn (ξ3′ ) u′ j
n
(14)
n=0
In the figures and the equation (14) the local directions are denoted as x′j , j = 1, 2, 3, while the global ones as xi , i = 1, 2, 3. In the case of the third local direction we also introduce the auxiliary dimensionless coordinate ξ3′ = 2x3′ /t, where t is the thickness of the shell. The corresponding nth mid-surface dof in the jth local direction is d ′ nj , while the nth through-thickness dof in the local direction j and the global direction i are u′ nj and uni , respectively. Note that the numbering of dofs in both cases is n = 0, 1, 2, ..., I, where I is the order of the transverse displacement field, equivalent to the order of the applied shell theory. More details on the 3D-based approach can be found in our works [13, 21].
2.1.2 Hierarchical modelling Our methodology utilizes the 3D-based adaptive hierarchical models M for complex structures, including the first-order Reissner-Mindlin (RM) shell theory, the hierarchical higher-order shell theories MI of order I, the three-dimensional theory of elasticity (3D), the solid-to-shell transition model (3D/RM), and the shell-to-shell transition models (MI/RM) as well. The complete set of the models is defined as:
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
x1
1
top
x'3
u' 3j f1u' 1j
u' 2j
x3
top
u3n
u'3n middle
u1n
ξ' 3
thickness
thickness
u' 2n
f 3u' 3j
middle
f 0u' 0j + f 3u' 3j
0
f 0u' 0j u' j (ξ' 3 )
x'2
f 2u' 2j
u2n
bottom u'1n
11
x2
u' 1j
-1
bottom
u' j
u' 0j
x'1
Fig. 8 Through-thickness dofs (left) and the displacement field (right) of the 3D-based approach
M ∈ M,
M = {3D, MI, RM, 3D/RM, MI/RM}
(15)
The subset MI of the hierarchical shell models as well as the corresponding subset MI/RM of the transition models are MI = { M2, M3, M4, ...}, MI/RM = {M2/RM, M3/RM, M4/RM, ...}
(16)
The elements of our hierarchy can be ordered with respect to the order I (or J), I = 1, 2, 3, ..., ∞ of the shell theory , i.e. M = RM ⇒ I = 1, J = I M = M2/RM, M3/RM, M4/RM ⇒ I = 2, 3, 4, ..., J = 1 M = M2, M3, M4, ... ⇒ I = 2, 3, 4, ..., J = I M = 3D/RM ⇒ I → ∞, J = 1 M = 3D ⇒ I → ∞, J = I is
(17)
The fundamental property of our hierarchy of the 3D-based mechanical models (18) lim lim kuI/J(M) kU,V = ku3D kU,V , J=1,I
I→∞
where the global norm of the solutions uI/J(M) , equal to the strain energy U within the body volume V , is kuI/J(M) kU,V =
1 2
Z
V
σ T (uI/J(M) ) ε (uI/J(M) ) dV,
(19)
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Grzegorz Zboi´nski
with σ and ε being the vectors of stresses and strains, respectively. Note that the subsequent solutions of the models of the hierarchy tend in the limit to the solution u3D of the model of 3D-elasticity, i.e. the highest model of the hierarchy. More information on the definitions of the models and the properties of our hierarchy can be found in our works [21, 13].
2.1.3 Hierarchical hpq-approximations For each of the mentioned models we introduce the two-dimensional, three-dimensional, or mixed (corresponding to the shell, solid and transition models, respectively) hpq-adaptive approximations, where h is the characteristic dimension of an element, while p and q represent longitudinal and transverse orders of approximation within the element, respectively. In other words, the subsequent solutions uI/J(M) to our subsequent models M of the hierarchy M are now approximated with the solutions uq(M),hp , in which the transverse order of approximation q is equivalent to the order I of the hierarchical model M, i.e. q ≡ I. Taking into account the specific character of the approximations for the cases of the first-order shell, hierarchical shell, and 3D-elasticity models, we can denote the specific solutions, corresponding to these models, in the following way M = RM, I ≡ J = 1 ⇒ uq(M),hp = uhp M ∈ MI/RM, I ≥ 2, J = 1 ⇒ uq(M),hp = uhpq/hp M ∈ MI, I ≡ J ≥ 2 ⇒ uq(M),hp = uhpq M = 3D/RM, I → ∞, J = 1 ⇒ uq(M),hp = uhpp,hp M = 3D, I ≡ J → ∞ ⇒ uq(M),hp = uhpp
(20)
In the above notation we simply skip q = 1 for the first-order shell model, and assume q = p for the model of 3D-elasticity. In the case of the transition models we perform accordingly. The evolution of the concept and the corresponding definitions of the hierarchical hpq-approximations for the structures of complex mechanical description can be investigated in our subsequent works [19, 16, 21, 13]. The main feature of the introduced approximations is that, with the increasing p and the increasing inverse of h, the numerical solutions tend in the limit to the exact solution uI/J(M) of the corresponding model M, i.e. lim kuq(M),hp k
1/h,p→∞
S e
U, V
= kuI/J(M) kU,V
(21)
Combination of the proposed hierarchy of mechanical models and the hierarchical approximations for these models leads to the hierarchy of adaptive numerical models for complex structure analysis (see [13]). The useful property of the solutions within this hierarchy is
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
lim lim lim kuq(M),hp k
J=1,I I→∞
1/h,p→∞
S e
U, V
= lim
J=1,I
13
lim kuI/J(M) kU,V = ku3D kU,V ,
I→∞
(22) which means that with the increasing discretization parameters p and 1/h and the increasing order I of the hierarchy of mechanical models we reach the solution corresponding to the exact solution of the highest model of the hierarchy.
2.1.4 Hierarchy of the hp-, hpq- and hpq/hp-adaptive finite elements The hierarchy of the numerical models, introduced in the previous subsection, is encoded into a hierarchy of the adaptive prismatic finite elements. The hierarchy includes: the hpp solid element, acting also as the hpq hierarchical shell elements, a family of the hpp/hp solid-to-shell transition elements, acting also as the hpq/hp shell-to-shell transition elements, and the hp first-order shell element. The local (element-level) model-, h-, as well as p- and q-adaptive capabilities of the hierarchy of elements results from the application of: the hierarchy of 3D-based mechanical models, the constrained approximation (hanging nodes) idea, as well as the hierarchical shape functions associated with the corresponding incremental degrees of freedom, respectively. These three ideas are presented in detail in our work [19]. It should be mentioned here that our works on the constrained approximation and hierarchical shape functions were inspired and took advantage of the works by the predecessors [7, 8]. Application of these concepts on the element level is elucidated in our work [20] for the solid and hierarchical shell elements, [23] for the transition elements, and [21] for the first-order shell element. The hierarchical character of the all elements is explained in [19, 22]. As the all above aspects of the assessed methodology are documented very thoroughly in the all mentioned works, we skip the corresponding details in this presentation.
2.2 Error estimation The applied error-controlled model- and hpq-adaptivity is based on the estimated values of the modelling and approximation errors, obtained from the residual equilibration method (ERM) [2, 3]. The method is applied twice, firstly for estimation of the approximation error and secondly for the modelling or total error. The method provides the upper bounds of the global approximation error for all the applied models as well as the upper bound of the global modelling error for the hierarchical shell models (thus also the upper bound of the global total error for the latter models can be proved). In the case of the first-order shell elements, and the corresponding solidto-shell transition elements, one cannot prove the upper bound of the modelling and total errors. As an undesired consequence only the global modelling error indicator can be obtained from the proposed approach. In the case of 3D elasticity model the
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Grzegorz Zboi´nski
total and approximation errors are equivalent and the upper bound of the errors can be shown. The starting point for the equilibrated residual method is the variational formulation in which the global error functional in the energy norm is employed. The error energy norm is defined as the strain energy of the difference of the exact and numerical solutions. Then, the functional is decomposed into the local (element-level) functionals for each finite element. In this step the upper bound property can be proved, based on the observation that the sum of the minimized potential energies from the decomposed local (element-level) variational problems is always grater than the global minimized potential energy of the body as a whole. As a consequence, the global strain energy of the error, for the whole body, is always grater than the sum of the strain energies of the errors from the decoupled local problems. Finally, we employ the global strain energy as an exact value of the error, and the element sum of the errors as the estimator to this exact error. This way the element residual approach to error estimation is formed. Details of our implementation of the method can be found in [19] for the whole hierarchy of our adaptive finite elements, and in [17, 18, 14] for the first-order shell element. Note that, from the implementation point of view, the obtainment of the solutions to the local (element-level) variational problems is very important, as these solutions contribute to the global estimator of the error. The local problems to be solved can be presented in the following form e
e
Q(M),HP Q(M),HP ,v )− L(vQ(M),HP ) − B(u
Z
e
T e
e
vQ(M),HP hr(uhp )i dS = 0,
(23)
S\(SP ∪SD )
e
e
where the bilinear B and linear L forms represent the virtual strain energy and the virtual work of the external forces of the body, both restricted to the element e. The solution and trial functions from the proper space of the element kinematically admissible displacements are denoted as uQ(M),HP and vQ(M),HP , with H, P, and Q standing for the element size, and longitudinal and transverse orders of approximation in the discretized local problem. The last left-hand side term of (23) represents the virtual work of the interelement stress reactions. The equilibrated version of e these reactions is denoted as hri. The equilibration means that the external and inef
ternal forces of the element are in equilibrium, and this is done with the vectors α of the splitting functions, to be determined on the common faces of the element e and any adjacent element f . Our definitions of these functions can be found in [2, 3, 19]. The formal definitions of the terms of the above relation are Z
e
Q(M),HP )= B(uQ(M),HP, v
e
e
e
T e e
e
e
T e
e
T e
ε T (vQ(M),HP )Dε (uQ(M),HP ) dV = v Q(M),HP k q Q(M),HP,
V
e
Q(M),HP
L(v
)=
Z
v
e
Q(M),HP T
V
e
f dV +
Z
T
e
e
v Q(M),HP p dS = v Q(M),HP (fM + fP ),
SP
Z
e
S\(SP ∪SD )
v
Q(M),HP T e
e
hr(uq(M),hp )i dS = v Q(M),HP fR
(24)
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
15
These terms can be also expressed in the language of the finite elements, by means e
of the element stiffness matrix k (defined with the elasticity matrix D and the strain e
e
e
vector ε ), and the element forces vectors fM , fP , and fR due to the body, surface, e and element reaction loads f, p, and hri, respectively. Denoting the searched nodal e Q(M),HP displacements as q , the relation (23) can be equivalently written as e e
e
e
e
k q Q(M),HP = fM + fP + fR
(25)
We will take advantage of the above form of the local problems later on in this chapter of the book.
2.3 Adaptive strategy Our adaptive strategy is based on Texas three-step strategy [9]. The original strategy lies in solution of the global problem thrice, on the initial, intermediate, and final meshes. The intermediate and final meshes are obtained through the local hrefinements (h-step) and local p-enrichments (p-step), respectively. In principle, the method leads to the solution of the assumed accuracy, as the error level is related to the discretization parameters through convergence theories. The original approach is enriched by us with the possible two or three iterations within the h- and p-steps. Additionally, we introduce one additional step proceeding the h- and p-steps, called modification one (see Fig. 9), in order get rid of such unpleasant phenomena as: • the improper solution limit of 3D elasticity model for q = 1 and thin structures, • numerical locking (for low values of p and 1/h within thin structures), • and boundary layers. In our additional step some global modifications (the global change of the model, the global p-enrichment, and the introduction of exponential mesh subdivision in the direction normal to the boundary) within the initial mesh are performed. This additional step needs special tools for detection of the mentioned three phenomena. The proposed tools take advantage of the algorithms of the applied error estimation method. They allow qualitative assessment of two local solutions obtained for the models for which these phenomena exist or not in accordance with the theory.
2.4 Possibility of other approaches One should be aware that the proposed approach is not the only option in adaptive analysis within computational solid mechanics. Within the modelling and approximation methods we can mention the methodologies based on one mechanical model, either the first-order shell [4], higher-order shells [1, 5], or 3D elasticity
16
Grzegorz Zboi´nski Start Reading data i=1 Initial mesh generation
Formation and solution of the problem equations Detecion of three phenomena (i=1) or error estimation (i=2,3,4) Final mesh generation Initial mesh modification Intermediate mesh generation Yes Yes
One of three phenomena detected?
i=2 i=1
i=2
i=3
No
i=2
No
i=2
No Yes
No
i=3
i=3 No
No
Enough intermediate error iterations? Yes
No
Enough final error iterations?
Yes
i=4
Yes
Target error achieved? Yes
Printing results Stop
Fig. 9 Adaptivity control with four-step iterative strategy
[11, 12]. Possible alternatives concerning the error estimation and adaptive strategies are: the goal-oriented adaptivity [10] and the so called automatic hp-adaptive strategy based on the two-mesh paradigm [6, 12], respectively. Even though, these methodologies were developed as an answer to the problem of poor effectiveness of other techniques, they do not answer to all questions concerning the adaptive modelling and analysis within solid mechanics. For example, the application of the automatic hp-adaptive strategy to the problems of complex mechanical description and problems with boundary layers is still an open question. Also, the goal-oriented approach, useful for practical applications, still does not give any general answer to the problem of upper and lower boundedness of the errors.
3 Chosen problems to be resolved and some remedies In this section we would like to present some chosen unresolved problems within the assessed methodology. Before the presentation of the problems and the related remedies we would like to state what follows.
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
17
• Our choice of the problems to be presented is rather subjective. We consider difficulties faced by us during implementation of our specific methodology. • The order in which we present these problems corresponds to the subsequent steps of the algorithm, not to the significance of the problems. • We search for the solution of these problems within the presented methodology, taking advantage of the potential hidden within the applied techniques. We do not consider the escape to other techniques (we do not want to introduce other quantities of interest, apply the goal-oriented adaptivity, or take advantage of the two-mesh paradigm). The short list and the characterization of the main problems, we would like to address in this chapter, is as follows. • Geometry modelling within 3D-based hierarchy of models is important because of its influence on the orthogonality of the modelling and approximation errors. • Excessive growth of the number of dofs in 3D can appear (especially in the case of corner or edge high solution gradients – they may need application of the 2-irregular constrained approximations). • Our approach does not give formal upper bounds for the free and forced vibration problems, however the effectiveness is practically identical as in the case of the equilibrium problem. • Poor or worse effectivity of the ERM estimator can be observed in the case of elongated elements and the meshes of locally varying order and size (the uniform meshes provide higher error but better effectivity than the adapted ones). • The changing convergence rates disturb the obtainment of the target error.
3.1 Hierarchical modelling issues In the context of the hierarchical modelling, we chose the specific questions of the robust formulations of the hierarchical shell models and transition models as well. One of the key issues within our 3D-based formulation is geometry modelling based on the middle surface and thickness concepts. Such geometry modelling is necessary for orthogonal decomposition of the total error into the modelling and approximation error components. The problem of the shell geometry modeling is illustrated in Fig. 10, where two systems of local coordinates are introduced. The first one corresponds to any arbitrary point of the shell, and is based on the assumption that the two local directions, determined by the vectors w1 and w2 , are tangent to the mid-shell surface, or to any other shell surface determined by the shell natural coordinates ξ1 and ξ2 , and additionally that the third direction, determined by w3 , is perpendicular to the mentioned surface. The second system of coordinates, corresponding to any point j of the lateral boundary of the shell, is defined in a different way. Here, the vector w3j is defined first, so as to coincide with the direction x3′ , i.e. to come through the corresponding points of the top and bottom surfaces of the shell. The surface determined
18
Grzegorz Zboi´nski
any point of the shell
x3
point j
ξ3 w
top surface
ξ3
x'3
j 3
w3 x'2
w 1j
w2
w 2j
x2
ξ1
x1
ξ2
w1 x'1
bottom surface
ξ2
shell mid-surface
ξ1 Fig. 10 The proper shell geometry definition
with the two remaining vectors, w1j and w2j , is now defined as perpendicular to the first vector. Note that when the first system of coordinates is generated in the point j, both the systems are not identical, unless the mid-shell surface is perpendicular to the direction x3′ . There is no problem with fulfillment of this requirement in the case of analytical description, provided that the shell geometry is defined as the one of the symmetric thickness in the direction perpendicular to the mid-surface. Note, however, that in the finite element approximation, the interpolated (polynomial) representation of the shell mid-surface may not be perpendicular to the lateral boundary of the shell. Fortunately, the higher the interpolation order, the better coincidence between the directions x3′ and ξ3 at any point j can be observed. Summing up, if one wants to take advantage of all the features of the shell theories, one has to construct geometry of the 3D shell body so as the lateral boundary of the shell is perpendicular to its mid-surface. In the finite element approximation of such shells, careful interpolation of their geometry is required.
3.2 Problems within hp-approximation What makes the implementation of the hp-adaptive approach difficult is the constrained approximation necessary for the local (element) h-refinements. In the case of mechanical systems of the complex mechanical description, the transition models and approximations make the issue even more complicated. Because of that below we discuss the chances of getting rid of these two techniques. In this subsection we
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
19
also address the problem of the excessive growth of the dof number in the case of 3D meshes.
3.2.1 The constrained approximation In this case we deal with the broken elements equipped with the so called hanging nodes. The general answer to the question if one can get rid of the hanging nodes is no. In the general h- or hp-approach, based on element refinement (as opposed to the remeshing technique), it is not possible. Note that the specific approach called Rivara’s refinement, where the division by four or two is led through the existing vertices of the rectangular faces of the elements, is applicable only to such faces. The approach requires the implementation of two types of elements within one mesh, with the rectangular and triangular faces in the undivided and divided elements, respectively. Note also that the original Rivara’s approach can be applied once. This is because, when one wants to repeat the division, he deals with the triangles, and the avoidance of hanging nodes requires the division by two of the element we want to divide and the adjacent element as well. In other words, in the case of triangular faces such an approach does not work unless we divide also the adjacent element. This makes things much more complicated and leads to over-divided meshes.
3.2.2 Necessity of the transition approximations In the case of the transition approximations, resulting from the application of the transition models joining the basic models together, the transition elements corresponding to such transition models are necessary. As an example of such a situation within solid mechanics we can mention the solid-to-shell transition elements acting between the 3D-elasticity model (or the higher-order shell models) and the firstorder Reissner-Mindlin shell model. The general answer to the question if one can get rid of the transition elements is no, again. There exist, however, some prosthesis approaches. For example, in the case of the bending-dominated shells, the ReissnerMindlin model can be replaced by the 3D-elasticity model of the transverse order q = 1, with changed elastic constants. In the case of the membrane-dominated shell structures, the Reissner-Mindlin model can be changed to the 3D-elasticity model of q = 1, without any modifications of the elastic constants. In both cases, the transition model and the corresponding transition elements are not necessary. The problem with such approaches is that, in the majority of technical applications, the character of the strain dominance (either bending or membrane one) is not known a priori. There are also some structures where a balance between the bending and membrane strains exists. Then, the only correct model is the Reissner-Mindlin one, which unfortunately requires the transition model, when combined with the 3D-elasticity one. As we search for a general approach to hierarchical modelling, we will not consider here the imperfect approaches mentioned above, suitable for the specific states of strains only.
20 p,q
Grzegorz Zboi´nski 28645883 2884 2874 2343 28545863 2183 5903 5843 2323 2223 2383 2203 2363 2023 2663 2163 862 6023 5683 2063 5823 2643 2043 2703 2503 2003 852 3574 3264 2683 5703 5663 3184 6063 6043 2483 3604 2543 3254 3584 3204 3284 3194 2523 25445523 692 652 3594 3504 3104 6003 5643 3274 2564 36633174 3494 2554 702 5503 3124 3524 662 5183 6603 4064 3683 3114 2743 2534 2704 332 2783 3514 2624 863 370336233094 5543 5483 2763 5223 6623 2694 2724 2723 382 2644 4084 3603 5203 3643 2634 342 463 4054 2464 883 5163 732 5463 2714 4074 5043 372 3823 32232614 1264 782 3843 2484 4143 3243 2474 50235003 5443742 3863 1424 1284 5063 326331832454 1254 4144 963 3463 4183 3163 823 3203 1434 1444772 3803 4983 4123 141412741184 4163 2863 33834803 4164 1204 903 1194 4963 39833403 4154 1174 4134 4783 3423 6743 4003 4023 6463 4823 4643 4623 3003 6123 3363 2823 3963 6383 4663703 3704 4763 3023 2983 6443 6143 1474 2963 4603 2843 3724 6363 763 5303 6543 3714 3694 6283 1803 1783 1723 1703 2803 5283 6523 6163 1823 1863 1743 6843 4463 4303 1763 1903 1683 1883 1843 6303 6183 1643 1623 6823 4503 4343 1663 4483 4443 4323 4283 1603
8
7
6
5
4
3
2
1
z x
y
Fig. 11 Edge and interior excessive growth of dofs number (the first mode of free vibration)
3.2.3 Growth of the dofs number in 3D problems There are at least three reasons for the excessive growth in the dofs number for 3D problems. • Application of only 1-irregular hanging nodes due to the constrained approximation. • Poor effectivity of the estimation (overestimation). • Wrong h- or p-convergence exponents in the adaptivity control procedures. Here we would like to address the first problem, while the remaining two will be discussed in the section concerning adaptivity control. So as to make the first problem less severe, we propose to extend the idea of the constrained approximation onto 2-irregular meshes. Note that the problem is especially important for large 3D structures with high edge or corner solution gradients. Application of 2-irregular meshes allow the avoidance of the excessive growth of dofs occurring when only 1irregular subdivisions are possible. Such a limitation in mesh generation causes that 1-irregular mesh penetrates deeply into the regions that could have been undivided, if a sequence of 1-irregular subdivisions had not been necessary due to solution continuity reasons. The example of the excessive growth of the number of dofs in the h-adapted, 1-irregular mesh, both in the interior and in the vicinity of the boundary, is shown in Fig. 11.
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
element f
21
element g
element e
2
2 1
1 2 1
2 1
element g
element f element e Fig. 12 1- and 2-irregular constrained nodes for vertical (left) and horizontal (right) subdivisions
The difference between the hanging nodes of 1- and 2-irregular meshes is illustrated in Fig. 12, where the node of the first type is marked with 1, while the node of the second type is denoted with the number 2. In order to retain solution continuity the displacements of the node 1 of the element f have to be equal to the interpolated e
displacements of the undivided, adjacent element e, at the location ξ 1 = 1/2 of the node 1 in the element e e f 1 e e q i1 = u i (ξ 1 ), ξ 1 = (26) 2 In the case of the element g, obtained through a double division, the displacements dofs at the node 2 can be expressed with either the interpolated displacements of the element e1 , that could have been obtained through a single division of the element e, or better directly with the interpolated displacements of the undivided element e, e
at the location ξ 2 = 3/4 of the node 2 in the element e g
e
e1
e
e
q i2 = u1 i (ξ 2 ) = u i (ξ 2 ),
e1 1 ξ 2= , 2
e
ξ 2=
3 4
(27)
As it can be seen, the procedure of the obtainment of the continuity conditions for the 1- and 2-irregular nodes is generally the same. As the interpolated displacements of the undivided element e can be expressed with the active displacement dofs of this element, then also the constrained dofs of the 1- and 2-irregular nodes can be expressed by them, and the corresponding constraint coefficient matrices can be defined. The general and detailed information on how to construct such matrices can be found in [8, 19, 14].
22
Grzegorz Zboi´nski
3.3 A posteriori error estimation In the error estimation problems, we would like to pay our attention to two issues. The first group of problems is theoretical and concerns the obtainment of the upper bound property of the residual estimators in the free and forced vibration problems. The second, implementation issue concerns poor or worse effectivity of the estimation.
3.3.1 Error estimation for the free and forced vibration problems In the free vibration problem the difficulty results from the fact that the exact solution in frequency (and in the corresponding strain energy) gives the lower bound of the numerical solution, opposite to the static case, where the exact solution in strain energy constitutes the upper bound. In the stationary forced vibration problem the main difficulty arises from the appearance of the phase angles in the acting forces. The solution to such a problem has to be obtained in the domain of complex functions. The phase angles have to be introduced into the formulation of the ERM estimators. Note that when these angles are equal to zero, the solution becomes real and the method suitable for the equilibrium problem can be applied directly. The similarities, differences, and the above mentioned theoretical difficulties within three analyzed problems of solid mechanics, in the frame of the presented methodology, can be summarized as follows. • The equilibrium problem: – the upper bound exists for the approximation, modelling and total errors (3Delasticty and 3D-based hierarchical shell models), and the approximation error (3D-based Reissner-Mindlin model). • The eigenproblem (free vibration): – no formal upper bound of the error can be proved, however effectivity of the estimation of the errors in the energy norm is practically the same as for the equilibrium problem (compare Fig. 13 and Fig. 14), – the main difficulty results from the lower boundedness of the numerical solution by the exact solution, opposite to the equilibrium (static) case, – the true error is the result of the error in the natural frequency and the error of the mode of vibration (the eigenvalue and eigenmode errors), – our methodology accounts only for the eigenmode error, – our methodology can be converted into the goal-oriented approach, with the normalized strain energy being the quantity of interest, and solutions of the ERM local problems forming the dual solution. • The forced-vibration problem: – the problem is solved for the amplitudes and phase angles – this requires the complex domain,
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
23
(X) 1.372 1.372
it
20 10
180 170 40 30
340 330 200 190 60 50
500 490 360 350 220 210 80 70
660 650 520 510 380 370 240 230 100 90
820 810 680 670 540 530 400 390 260 250 120 110
980 X 970 840 830 700 690 560 550 420 410 280 270 140 130
1140 1130 1000 990 860 850 720 710 580 570 x 440 430 300 290 160 150
max
1.208 1.167 1.063 1160 1150 1020 1010 880 870 740 730 600 590 460 450 320 310
avr
0.936 1180 1170 1040 1030 900 890 760 750 620 610 480 470
0.824 1200 1190 1060 1050 920 910 780 770 640 630
0.726 1220 1210 1080 1070 940 930 800 790
0.639 1240 1230
0.563 1260 1250 0.496 1100 1280 1090 1270 1120 0.436 1110 960 0.384 950 0.338 0.298 0.262 0.231 0.203 (x)
0.179 0.179
min
z x
y
Fig. 13 Total error effectivities for the equilibrium problem
– there is still no formal answer to the question of the upper boundedness of the errors (the work is in progress), – one can expect the effectivity of the estimation to be similar as for the equilibrium and free vibration problems (to be demonstrated numerically).
3.3.2 Effectivity of the estimation The second issue has an implementation character and deals with the poor effectivity of the residual approach in the case of elements elongated in order to resolve the boundary layers. The corresponding exemplary effectivities for the regular meshes of the same degrees of freedom, with square and elongated elements, are presented in Fig. 15 and Fig. 16, where the global effectivities are 1.2 and 11.5, respectively. Worse effectivity appears also for the locally hp-adapted elements, in comparison with the uniform elements. The meshes to be compared, of the similar number of dofs and the same uniform order of approximation, are presented in Fig. 15 and Fig. 17. The global effectivities are about 1.2 and 1.7, respectively. One of the available remedies, available but not implemented yet in 3D, is the application of the equilibration of the higher order. Such a method might include the equilibration not only at the vertices of the elements. For the first case, when only the vertex equilibration is performed, the directional components i = 1, 2, 3 of
24
Grzegorz Zboi´nski (X) 1.774 1.774
it
max
1.551
180 170
20 10
40 30
340 330 200 190 60 50
500 490 360 350 220 210 80 70
820 810
660 650
680 670
520 510
540 530
380 370
400 390
240 230
260 250
100 90
120 110
1140 1130 X
980 970
1160 1150
1000 990
840 830
1020 1010
860 850
700 690
880 870
720 710
560 550
740 730
580 570
420 410
600 590
440 430
280 270
460 450
300 290
140 130
320 310
160 150
1.356 1.188 1.185 1180 1170 1040 1030
avr
1.036 1200 1190
900 890 760 750 620 610 480 470
1060 1050 920 910 780 770 640 630
0.906 1220 1210 1080 1070 940 930 800 x 790
0.792 1240 1230
0.692 1260 1250 0.605 1100 1280 1090 1270 1120 0.529 1110 960 0.462 950 0.404 0.353 0.309 0.270 0.236 (x)
0.206 0.206
min
z x
y
Fig. 14 Total error effectivities for the first mode of free vibration
the vectors of splitting functions can be expressed through the six (in the case of the applied prismatic elements) vertex splitting coefficients. For the higher-order nodes no equilibration is performed, and the splitting coefficients, equal to 1/2, reflect the averaging of the nodal reactions between the elements. This type of equilibration can be characterized with the relation ef
αi=
6
∑
j=1
ef e 1e α ij χ j + ∑ ∑ χ j,m , j>6 m 2
(28)
e
where χ j represents the shape function of the vertex node j. Note that the non-zero contributions to the splitting functions correspond to four out of six vertex nodes, located on the common face of the elements e and f (see Fig. 18). In the higher order equilibration also the higher order mid-edge and/or mid-side nodes are included in the equilibration procedure (see [2, 19]). The respective definition of the splitting function components are ef
αi=
6
∑
j=1 e
ef
e
ef
e
α ij χ j + ∑ ∑ α ij,m χ j,m ,
(29)
j>6 m
with χ j,m standing for the shape function corresponding to the higher-order node j, and dof m, defined in this node. Now, the non-zero contributions to the face splitting
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
25
(X) 6.287 6.287
ia
max
5.267 4.412 260
3.697 3.097
250 180
280
170 100
2.594
270 200
2.174
300
1.821 90 20
190 120
10
110 40
290 220 210
140
30
320 1.526
240
130 60
310X
230
1.278 1.205 1.071
avr
0.897
160
0.752 50 x
150 0.630
80
0.527
70
0.442 (x)
0.370 0.370
min
z x
y
Fig. 15 Effectivities for the uniform mesh
function correspond to four vertices, four mid-edge nodes and the mid-side node of the common face The presented idea looks very simple but our numerical tests show that the effective higher-order equilibration is not a trivial task. The results can be even worse than in the case of linear equilibration. The second suggested remedy for poor or worse effectivity of the estimation can be the constraining of the element displacement field while solving local problems of the ERM. This approach is not very common and it needs further theoretical and numerical studies.
3.4 A posteriori detection of the undesired phenomena The question on how to detect the improper solution limit, the numerical locking and the boundary layers a posteriori is still another problem. The first phenomenon leads to the incorrect solution, while the latter two cause varying convergence rates and disturb the desired exponential convergence. Also the problem of coping with these phenomena via automatic, error-controlled, adaptive approach is still open. Below we present our numerical tools [15] for the detection of the mentioned phenomena.
26
Grzegorz Zboi´nski (X) 33.069 33.069
ia
max
23.115 260 250 180 280 170 200 x270 190 100 300 120 220 90 290 110 140 210
16.158 11.545 11.295
130
7.895
20 40
5.519
320 240
60
avr
160
3.858
1030
310 2.697 230 50 X
80
150
1.885 1.318 0.921 0.644
70
0.450 0.315 0.220 0.154 (x)
0.107 0.107
min
z x
y
Fig. 16 Effectivities for the non-uniform mesh
These tools take advantage of the algorithms of the residual equilibration method applied to error estimation.
3.4.1 The improper solution limit and numerical locking The improper solution limit and the locking is illustrated in Fig. 19. The first phenomenon appears in thin structures modelled with the 3D model, when the transverse approximation order equals unity (q = 1). Then, the solution is the fraction of the correct solution, represented by the value of 1 in the figure, regardless the structure length to thickness ratio t/l. Note that in the figure the relative value of the solution is calculated as the strain energy U, related to the reference, analytical value Ur of this energy. The numerical locking occurs in thin structures when the longitudinal approximation order is low (p = 1 in the figure) and/or the mesh density is not high (number of longitudinal edge subdivisions m equals 4 in the figure), regardless the value of the transverse approximation order q. In such situations, the locked solutions tend to zero in the thin limit. Our numerical tool for the detection of the improper solution limit, lies in the comparison of two solutions obtained from the local problems for the element e chosen from the interior of the thin structure (see Fig. 20). The first solution is
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
27
(X) 8.628 8.628
ia
max
7.326 6.221 114
5.283 4.486
124 94 174 154
204 184
164 134
104 194
294
354 364
334 254
344 274
284 264
144
324 304
3.810 314 3.235
234
2.747
244 214
224 x
2.333 1.981
22
34
12
42 X
32
1.682 1.658 1.428
44 14
24
avr
1.213 1.030
30
0.875 0.743 (x)
0.631 0.631
min
z x
y
Fig. 17 Effectivities for the hp-adapted mesh
characterized with the model and discretization data for which the phenomenon may appear, and the second one with the parameters for which the phenomenon does not appear, i.e. e e
e
e
e
e e Q(RM),HP
e
e
e
k q Q(3D),HP = fM + fP + fR , kq
= fM + fP + fR ,
Q = 1,
P = 8,
Q = 1,
P=8
(30)
The analogous detection strategy is applied to the locking phenomena, either the shear or membrane ones (for the explanation see [19]). The local solutions to be compared are now obtained from the following problems e e
e
e
e
e e Q(M),HP
e
e
e
k q Q(M),HP = fM + fP + fR , kq
= fM + fP + fR ,
P = p, P=8
(31)
Note that the same approach can be also utilized for the assessment of the intensity of locking. For this purpose we perform a kind of sensitivity analysis with the changing value of the longitudinal order of approximation P. The information on how to interpret the results from the local problems, concerning the detection of the numerical locking and the improper solution limit, and the
28
Grzegorz Zboi´nski
element f v4
v4 e3
element e v3
v3
e4 s v1
e2
v1
e1
element f v2
v2
element e Fig. 18 Linear (left) and higher-order (right) equilibrations 5.0E+0
MI, m=4 q=2, p=1 4.0E+0
q=2, p=4,5,6,7,8 q=1, p=4,5,6,7,8
U/Ur
3.0E+0
2.0E+0
1.0E+0
Fig. 19 Illustration of the improper solution limit and numerical locking
0.0E+0 1.00
10.00
100.00
1000.00
l/t
assessment of the intensity of the locking, can be found in [19, 15]. The main difficulty in this interpretation consists in the dependence of the obtained local solutions on the element dimensions resulting from the density of the global mesh (the quality of the detection is better for dense meshes).
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
29
element e
Fig. 20 Single-element local problem for the detection of the improper solution limit and numerical locking
3.4.2 The boundary layer phenomenon The phenomenon appears in thin and thick structures, when the analytical solution is the sum of the smooth part, corresponding to the interior of the analyzed domain, and the boundary part of high gradients. In the numerical solution of such problems, the interpolation functions suitable for the smooth part of the analytical solution, may not be appropriate for modelling solution gradients in the vicinity of the boundary. The situation is illustrated in Fig. 21, where the convergence curves for three cases are presented. The curves relate the numerical error obtained as a difference of the numerical and reference (exact) values of the strain energy, U and Ur , with the number N of degrees of freedom within the numerical model. The first case corresponds to the plate problem described with the Reissner-Mindlin model (q = 1). This model is not very much prone to the boundary layer phenomenon and the corresponding convergence is very high. Note that the uniform mesh of the division number m = 8 is applied in this case. The second case corresponds to the same plate and the model changed to 3D-elastic one. We apply the same uniform mesh of m = 8. It appears that the convergence is very poor in comparison to the previous case. The reason is the application of the 3D model of the transverse approximation order q = 2, very much prone to the edge effect. In order to resolve the problem of the poor convergence, we introduce the mesh of constant density in the interior of the plate, and varying density in the part adjacent to the boundaries, where the exponential subdivisions towards the boundary are applied. This mesh is denoted with the division number m = 4 + 4. The application of this mesh restores the high convergence of the solution, as it can be seen from the third presented curve. Our numerical tool, for the detection of the boundary layers, is based on the comparison of two solutions obtained from the local problems for the chosen pair of elements adjacent to the structure boundary (see Fig. 22). The first solution is obtained from the problem characterized with the uniform subdivision of the pair of elements into four smaller elements. For such a subdivision the edge effect may
30
Grzegorz Zboi´nski 3.0 p=1
p=1 p=2
2.0 p=2
p=2
1.0 p=3 p=4 p=3
log (U-Ur)
0.0
p=3
-1.0
-2.0 p=4
t/l=0.33%
-3.0
RM, q=1, m=8
p=4
3D, q=2, m=8 -4.0
3D, q=2, m=4+4
-5.0
Fig. 21 Convergence affected by the boundary layer
2.0
2.5
3.0
3.5
4.0
4.5
log N
appear. In the second problem we apply the exponential subdivision (see Fig. 22, again) and we can expect the solution to be free of the edge effect. Our local problems to be solved are 4
4 fi f i
fi
fi
fi
i=1 4 fi
fi
fi
∑ k q Q(M),HP = ∑ ( f M + f P + f R ), Hn,i = h/2, i = 1, 2, 3, 4,
i=1 4 fi f i Q(M),HP
∑ kq
i=1
=
∑ ( f M + f P + f R ), Hn,1 = Hn,2 = h/10, Hn,3 = Hn,4 = 9h/10,
i=1
(32) where Hn,i , i = 1, 2, 3, 4 represent the mesh dimensions of the four smaller elements in the direction normal to the boundary. The element dimensions in the direction tangent to the boundary are kept unchanged and equal Ht = h, with h standing for the dimension of the chosen pair of elements. We would like to add that the similar approach can be utilized for the assessment of the intensity and range of the phenomenon. For this purpose, the performance of a kind of sensitivity analysis is necessary, with the changing value of the elements dimension Hn,i (in the normal direction) in the exponential subdivision of the chosen pair of elements. The information on how to interpret the results from the local problems for the detection and the assessment of the intensity and range of the phenomenon are pre-
Unresolved problems of hierarchical modelling and hp-adaptive analysis...
31
element f4 element f3
element f2
element f1
Fig. 22 Four-element local problem for the detection of the boundary layer phenomenon
sented in [19, 15]. The main difficulty in this interpretation is the same as for the detection of the improper solution limit and the numerical locking (the dependence on the global mesh density). The additional difficulty is that in the case of the exponential subdivisions we deal with the elongated elements, and the quality of the boundary layer detection worsens.
3.5 Adaptive procedures Finally, we will address the adaptive strategy issues. In particular, we will discuss the problem of changing convergence rates, which makes achieving the target admissible value of the error difficult within the original three- or our four-step strategies. Having a closer look at the p- or h-convergence curves (Fig. 23 and Fig. 24, respectively) of the numerical solutions to the problems subject to the numerical locking and boundary layers, one can distinguish three different regions. The first region corresponds to the numerical locking (the horizontal parts of the curves in both figures), the second one to the exponential or algebraic convergence (the middle parts of the curves), and the third region to the lost of regularity due to boundary layers (the third parts of the curves). Though, for each of these three regions some convergence theories exist, the transition from one state to another is unclear and difficult to be determined analytically as a function of the structure thickness t and the discretization parameters h, p, and q. The problem is very similar to that mentioned in Sect. 3.2.3, where we dealt with the wrong or unknown values of the exponents of the convergence curves. Note that one of the remedies for the problems of this type can be the application of the corrective iterations within the h- and p-steps of the adaptive procedure. This idea has already been implemented in our algorithms and programs. The significance of the issue can be better understood when the standard relations (see [9, 19]) controlling the h- and p-adaptivity are taken into account. The convergence exponents µ0 and ν0 from the initial mesh, and the estimated values of the element error, η0 and ηI , from the initial and intermediate meshes, influence
32
Grzegorz Zboi´nski 2.00 MI, q=1, t/l=1% m=1
p=1 p=1
p=2 p=3
p=1
m=4
p=2
0.00
log (U-Ur)
p=4
m=7
p=2
p=5 p=3
-2.00 p=3 p=6 p=4
-4.00 p=5
p=4 p=6 p=5 p=6
Fig. 23 Changing pconvergence rates for thin structure problems
-6.00 1.00
2.00
3.00
4.00
log N
2.00 MI, q=1, t/l=1% m=1
m=2
m=4
m=6 m=8
p=1
m=1
p=3
0.00
p=5
m=2
log (U-Ur)
m=1
m=3 m=4
-2.00 m=6 m=8 m=2
-4.00
m=3 m=4 m=5 m=6 m=8
Fig. 24 Changing hconvergence rates for thin structure problems
-6.00 1.00
2.00
3.00
4.00
log N
very much the number nI of the intermediate mesh elements replacing the initial mesh element η 2 EI 2µ /d+1 nI 0 = 20 2 (33) γI ku0 kU and the value of the approximation order pT in the target (final) mesh
Unresolved problems of hierarchical modelling and hp-adaptive analysis... 2ν
pT 0 =
p02ν0 ηI2 EI , γT2 ku0 kU2
33
(34)
where d = 3 is the dimensionality of the 3D problem, γI and γT are the assumed, admissible, relative values of the intermediate and target (final) errors, and EI is the total number of elements in the intermediate and final meshes. Note also that the global strain energy norm of the global solution u0 from the initial mesh is utilized in the above equations.
4 Conclusions The main problems of the adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics have been presented. This has been done in the context of the proposed methodology, consisted of: the 3D hierarchical modelling, the hierarchical approximations, the a posteriori error estimation with the equilibrated residual method, and the four-step adaptive strategy. We have addressed such problems as: the geometry modelling within 3D-based thin structures, the over-divided meshes, the lack of the upper bound property for the free and forced vibration problems, the poor or worse effectivity of the estimation for the meshes with the elongated or non-uniform elements, the unsatisfactory quality of the detection of three undesired numerical phenomena, and the changing convergence rates for thin structures. The remedies for overcoming these problems have been indicated, basing on the described theoretical premises and our numerical experiments.
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