 Research article
 Open Access
On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity
 Sandrine Germain^{1}Email author,
 Philipp Landkammer^{1} and
 Paul Steinmann^{1}
https://doi.org/10.1186/22137467110
© Germain et al.; licensee Springer. 2014
 Received: 22 August 2013
 Accepted: 4 April 2014
 Published: 11 April 2014
Abstract
Background
Inverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought.
Methods
In this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the pathdependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached.
Results
In our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved.
Conclusions
With the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.
Keywords
 Inverse form finding
 Elastoplasticity
 Large strain
Background
In this work we present a recursive method for the determination of the undeformed configuration of a functional component, when only the deformed configuration of a workpiece, the applied forces and the boundary conditions are previously known. This is commonly known as an inverse form finding problem, which is inverse to the standard direct kinematic analysis in which the undeformed sheet of metal, the applied forces and boundary conditions are known while the deformed state is sought. Inverse form finding methods are useful because they allow to conceive designs at less time and at lower costs compared to an experimental approach.
Govindjee, 1996 and 1998[1, 2] proposed a numerical method for the determination of the undeformed shape of a continuous body, which is based on the work originally presented in[3]. Their work is limited to isotropic compressible neoHookean and incompressible materials. In these contributions it was shown that the weak form of the inverse motion problem based on the Cauchy stress is more efficient and straightforward compared to the weak form based on the Eshelby stress. The governing equation underlying the numerical analysis of the inverse form finding problem is therefore the common weak form of the balance of momentum formulated in terms of the Cauchy stress tensor. The unconventional result lies in the fact that all quantities are parameterised in the spatial coordinates. In[4], temperature changes in the undeformed and deformed configurations have been taken in consideration for orthotropic nonlinear elasticity and axisymmetry using a St.Venant type material, i.e., a material characterised by a quadratic free energy density in terms of the GreenLagrange strain. Koishi, 2001[5] used the previous method for the purpose of tire design. Yamada, 1998[6] proposed another approach as in[1] based on an arbitrary Lagrangian–Eulerian kinematic description. The arbitrary LagrangianEulerian description is approximated by a finite element discretisation. In the last decade,[7, 8] extended the method proposed in[1] for the case of anisotropic hyperelasticity for a St.Venant type material. This work is extended in[9, 10] to inverse analysis of largedisplacement beams in the elastic range. Lu, 2007[11] proposed a computational method of inverse elastostatics for anisotropic hyperelastic solids in the context of fibrous hyperelastic solids and provide explicit stress function for soft tissue models. In[12] an inverse method for thinwall structures modelled as geometrically exact stress resultant shells is presented. Germain, 2010 and 2013[13–15] extended the method originally proposed in[1] to anisotropic hyperelasticity that is based on logarithmic strains. This work was further extended to anisotropic elastoplasticity in[15, 16]. The authors demonstrated that the inverse mechanical formulation in elastoplasticity can be used only if the set of internal variables at the deformed state is previously given. However, when dealing with metal forming processes, this set of internal variables is not known at the deformed state. To overcome this problem in anisotropic elastoplasticity,[15, 17] proposed a numerical method based on shape optimisation in order to solve inverse form finding problems. A gradientbased shape optimisation is used in the sense of an inverse problem via successive iterations of a direct mechanical problem. The objective function is defined by a leastsquare minimisation of the difference between the target and the current deformed configuration of the workpiece. The design variables are chosen as the node coordinates stemming from the Finite Element (FE) formulation. A drawback of a nodebased shape optimisation is the possible occurrence of mesh distortions. Germain, 2013 and 2012[15, 18, 19] proposed a recursive algorithm using an update of the reference configuration. This proposal allows to avoid mesh distortions but leads to large computational costs. Germain, 2011 and 2012[20, 21] compared the inverse mechanical and the shape optimisation formulation in terms of computational costs and accuracy of the obtained undeformed functional component. They have shown that both methods lead to the same results, but the shape optimisation formulation has larger computational costs. In a similar way[22] dealt with the optimal design and optimal control of structures undergoing large rotations and large elastic deformations. Ibrahimbegovic, 2003[23] introduced shape optimization of elastic structural systems undergoing large rotations. Sousa, 2002[24] proposed an approach to optimal shape design in forging. In their recursive formulation the inverse problem is formulated as an optimisation problem, where the objective function sensitivity is calculated by the accumulated sensitivities of the nodal coordinates throughout the entire simulation of the process. Ponthot, 2006[25] presented optimisation methodologies for automatic parameter identification and shape/process optimisation in metal forming simulation. In the sensitivity analysis they used a disturbed balanced configuration, which is updated until the residual equilibrium of the disturbed problem ends under a fixed tolerance. Recently,[26] proposed an inversemotionbased form finding for electroelasticity to improve the design and accuracy in electroelastic applications such as grippers, sensors and seals.
In order to overcome the large computational costs ([20, 21]) in shape optimisation and the fact that the set of internal variables is unknown at the deformed state, we propose, in this contribution, a new method for solving inverse form finding problems in isotropic elastoplasticity based on the inverse mechanical formulation originally proposed in[1].
The present work is organised as follows: In order to introduce the utilised notations, the kinematics of geometrically nonlinear continuum mechanics are presented at first. Furthermore a macroscopical phenomenological isotropic elastoplastic model based on the additive decomposition of the total strains in the logarithmic strain space is introduced. A direct and an inverse mechanical formulations for determining the deformed and the undeformed configurations of a workpiece are respectively presented. A recursive formulation for solving the inverse form finding problem in isotropic elastoplasticity is developed using both previously presented formulations. To illustrate the proposed recursive formulation three numerical examples are presented. The influence of the starting values for the set of internal variables at the beginning of the computation is finally discussed.
Methods
Kinematics of geometrically nonlinear continuum mechanics
Nonlinear isotropic elastoplastic material model in the logarithmic strain space
The transposition symbol refers to an exchange of the first and last pairs of index.${\mathbb{E}}^{\mathit{\text{ep}}}$ is the fourthorder elastoplastic tangent modulus (see for example[32]).
Direct mechanical problem for determining the deformed shape from equilibrium
where (i,j) are the node numbers,$\stackrel{2}{\xb7}$ denotes the contraction with the second index of the corresponding tangent operator and$\overline{\otimes}$ denotes a nonstandard dyadic product with${[\mathit{A}\overline{\otimes}\mathit{B}]}_{\mathit{\text{ijkl}}}={A}_{\mathit{\text{ik}}}{B}_{\mathit{\text{jl}}}$. Due to the computation of the direct mechanical formulation the pathdependency, which has to be considered in elastoplasticity, is ensured.
Inverse mechanical problem for determining the undeformed shape from equilibrium
where (i,j) are the node numbers,$\stackrel{2}{\xb7}$ denotes the contraction with the second index of the corresponding tangent operator and$\underline{\otimes}$ denotes a nonstandard dyadic product with${[\mathit{A}\underline{\otimes}\mathit{B}]}_{\mathit{\text{ijkl}}}={A}_{\mathit{\text{il}}}{B}_{\mathit{\text{jk}}}$. For more details see[13] or[15]. Furthermore[15, 16] demonstrated that this inverse mechanical formulation might be used in elastoplasticity, when the set of internal variables at the deformed state is given and remains constant during the iterations. Since however in sheet metal forming processes the set of internal variables is not known at the deformed state, this formulation can not be used because the undeformed configuration will thus not be unique.
Form finding optimisation scheme

Find X such that$f(\mathit{X})=\frac{1}{2}{\mathit{\phi}(\mathit{X}){\mathit{x}}^{\text{target}}}^{2}\to \underset{\mathit{X}}{min}$

Subject to:
 1.
F = ∇_{ X }φ, C = F ^{ T }·F, $\mathit{E}=\frac{1}{2}ln\phantom{\rule{.3em}{0ex}}\mathit{C}$, E = E ^{ e } + E ^{ p }
 2., $\mathit{S}=\mathit{T}:\mathbb{P}$, P = F·S$\mathit{T}=\mathbb{E}:{\mathit{E}}^{e}$
 3.
Div(P) = 0, $\mathit{P}\xb7\mathit{N}=\overline{\mathit{T}}$, $\mathit{\phi}=\overline{\mathit{\phi}}$

and along trajectory x = x(t) with x(0) = X, ∀t:
 1., $\stackrel{\u0307}{\mathrm{\gamma}}\mathrm{\Phi}=0$ with Φ = Φ(T,α) and $\stackrel{\u0307}{\alpha}=\stackrel{\u0307}{\mathrm{\gamma}}\sqrt{\frac{2}{3}}$$\mathrm{\Phi}\le 0,\phantom{\rule{1em}{0ex}}\stackrel{\u0307}{\mathrm{\gamma}}\ge $
 2.$\stackrel{\u0307}{\mathrm{\gamma}}\stackrel{\u0307}{\mathrm{\Phi}}=0$
 3.${\stackrel{\u0307}{\mathit{E}}}^{p}=\stackrel{\u0307}{\mathrm{\gamma}}\frac{\partial \mathrm{\Phi}}{\partial \mathit{T}}$
Recursive formulation for solving inverse form finding problems
Remark:

If the set of internal variables is again set to a homogeneous field equal to zero and not updated to the heterogeneous field$({\mathit{E}}_{\text{current}}^{p},{\alpha}_{\text{current}})$ in the inverse computation the wanted undeformed configuration will not be reached.
Experiments and results
The internal variables are initialised to zero at the beginning of the computation. The convergence tolerance was set to ε = 10^{8}. Each numerical example was computed on an Intel Core2 Duo (2533 MHz).
Numerical example 1: bar
Material parameters
Material parameters  

E  211000 MPa 
ν  0.3 
h  100 MPa 
σ _{0}  415 MPa 
σ _{ ∞ }  750 MPa 
w  15 
Bar: Convergence
Bar: Calculation of Δ  

Iteration  {E^{ p }, α} = {0, 0}  {E^{ p }, α} = {0.1, 0}  {E^{ p }, α} = {0, 0.4} 
1  8.8532 10^{3}  53.7853  37.5161 
2  7.1592 10^{5}  2.3427  163.3108 
3  4.7781 10^{6}  6.3799  9.0531 10^{2} 
4  3.303 10^{7}  10.0643  1.4682 10^{2} 
5  1.4706 10^{8}  49.9705  1.4356 10^{5} 
6  4.7759 10^{10}  9.6875  1.2357 10^{5} 
7  29.0923  1.6002 10^{8}  
8  7.8441  1.2104 10^{8}  
9  20.5272  1.2903 10^{11} 
Numerical example 2: Cook’s membrane
Cook’s membrane: Convergence
Cook’s membrane: Calculation of Δ  

Iteration  {E^{ p }, α} = {0, 0}  {E^{ p }, α} = {0, 0.1}  {E^{ p }, α} = {0.02, 0.02} 
1  84.5661  3124.0123  2811.6881 
2  5.8472 10^{2}  9.5801  257.2319 
3  2.6557 10^{2}  2.8556  49.982 
4  2.9812 10^{3}  4.5033 10^{1}  61.4643 
5  2.4846 10^{4}  3.6374 10^{2}  102.2576 
6  1.6952 10^{5}  2.3578 10^{3}  229.8063 
7  9.859 10^{7}  1.4145 10^{4}  477.2713 
8  5.7253 10^{8}  7.924 10^{6}  1092.9303 
9  3.1401 10^{9}  4.4152 10^{7}  2632.0609 
10  2.4095 10^{8}  
11  1.3194 10^{9} 
Numerical example 3: circular, flat plate
Circular plate: Convergence
Circular plate: Calculation of Δ  

Iteration  {E^{ p }, α} = {0, 0}  {E^{ p }, α} = {0, 0.1}  {E^{ p }, α} = {0.1, 0} 
1  1.897 10^{2}  13.3091  88.913 
2  2.0153 10^{4}  5.9613 10^{3}  11.4455 
3  2.1253 10^{6}  1.8077 10^{4}  22.8091 
4  4.1797 10^{8}  1.3011 10^{6}  50.1605 
5  4.4199 10^{10}  4.3398 10^{8}  129.4614 
6  9.6423 10^{10} 
Discussion
In this section the influence in the choice of the starting value (initialisation of the recursive formulation) for the set of internal variables on the bar, the Cook’s membrane and the circular plate is discussed. We used for a more convenient implementation constant single numerical values for the starting set of internal variables instead of a non homogeneous field.
Influence of the starting values of the internal variables on the bar
The presented method was computed as in the previous section but this time the internal variables were first initialised to {E^{ p },α} = {0,0.4} and subsequently to {E^{ p },α} = {0.1,0}. It was found that the computation took 10 minutes 40 seconds when {E^{ p },α} = {0,0.4}. The convergence tolerance ε was obtained after nine iterations (Table2). The values Δ after each iterations are plotted in Figure4 (black curve).
i.e., the difference is negligible. In the case where {E^{ p },α} = {0.1,0} the computation was stopped after nine iterations because of the divergence of the Δ values. The Δ values after each iterations are plotted in Figure4 (red curve). We conclude that the initial values of the internal variables chosen for the computation of the recursive formulation have an influence on the computational costs and on the number of iterations needed to reach the convergence but not on the final result, i.e., on the sought undeformed configuration of the bar, if convergence can be achieved.
Influence of the starting values of the internal variables on the Cook’s membrane
The presented method was computed as in the previous section but this time the internal variables were first initialised to {E^{ p },α} = {0,0.1} and subsequently to {E^{ p },α} = {0.02,0.02}. It was found that the computation took 5 hours 23 minutes 34 seconds for the first case. The convergence tolerance ε was obtained after 11 iterations (Table3). The values Δ after each iterations are plotted in Figure7 (red curve). It can be observed that the rate of convergence is almost linear.
i.e., the difference is negligible. In the case where {E^{ p },α} = {0.02,0.02} the computation was stopped after nine iterations because of the divergence of the Δ values. The Δ values after each iterations are plotted in Figure7 (black curve). We conclude that the initial values of the internal variables chosen for the computation of the recursive formulation have an influence on the computational costs and on the number of iterations needed to reach the convergence but not on the final result, i.e., on the sought undeformed configuration of the Cook’s membrane, if convergence can be achieved.
Influence of the starting values of the internal variables on the circular, flat plate
The presented method was computed as in the previous section but this time the internal variables were first initialised to {E^{ p },α} = {0,0.1} and subsequently to {E^{ p },α} = {0.1,0}. It was found that the computation took 21 minutes 57 seconds for the first case. The convergence tolerance ε was obtained after six iterations (Table4). The values Δ after each iterations are plotted in Figure10 (red curve). It can be observed that the rate of convergence is almost linear.
i.e., the difference is negligible. In the case where {E^{ p },α} = {0.1,0} the computation was stopped after five iterations because of the divergence of the Δ values. The Δ values after each iterations are plotted in Figure10 (black curve). We conclude that the initial values of the internal variables chosen for the first computation of the recursive formulation have an influence on the computational costs, but not on the final result, i.e., on the sought undeformed configuration of the circular, flat plate, if convergence can be achieved.
Conclusion
In this contribution a new method for solving inverse form finding problems for isotropic elastoplastic materials is presented. To that end, a recursive formulation is deployed to find the desired undeformed configuration of the functional component. The inverse mechanical formulation in elastoplasticity is first performed on the target deformed configuration of the workpiece with the set of internal variables initialised to a homogeneous field equal to zero. Subsequently, a direct mechanical formulation on the computed undeformed configuration is used, which ensures the pathdependency in elastoplasticity. The obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the elastoplastic inverse mechanical formulation is started again with the target deformed configuration and the current heterogeneous state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached. Three numerical examples, a bar, the Cook’s membrane and a circular, flat plate in 3D illustrated this recursive formulation for finding the corresponding undeformed configurations in isotropic elastoplasticity. The convergence was reached after six, nine and five iterations, respectively, when initialising the set of internal variables to zero at the beginning of the computation. The influence of the starting values for the set of internal variables at the beginning of the computation was afterwards discussed. It was found that when the initial set of internal variables was initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. The rates of convergence were almost linear. The computation of the three numerical examples with the recursive formulation did not converge for one value of the set of internal variables and had to be stopped. Comparing the results of the numerical examples, it was demonstrated that different starting values for the set of internal variables have no influence on the obtained undeformed configuration. We conclude that the choice of the initial set of internal variables has an influence on the convergence evolution but not on the result, if convergence can be achieved. Therefore an initial homogeneous set of internal variables equal to zero, which is a natural choice in programming since the set of internal variables is unknown at the beginning of the computation, should be considered for such problems. An extension of the presented new method for solving inverse form finding problems to anisotropic elastoplasticity will be of great interest for metal forming processes.
Declarations
Acknowledgements
This work is supported by the German Research Foundation (DFG) under the Transregional Collaborative Research Center SFB/TR73: "Manufacturing of Complex Functional Components with Variants by Using a New Sheet Metal Forming Process  SheetBulk Metal Forming".
Authors’ Affiliations
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