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Multidisciplinary robust and reliability-based design optimization of injection molding system

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Abstract

Even a slight reduction in production cost can make a huge impact in a mass manufacturing domain like injection molding. In this paper, we proposed design modifications to the conventional multi-cavity injection mold insert for the reduction of overall mold material costs required for molding a part family. Additionally, the reduced size of the proposed insert makes it more suitable to manufacture it using metal additive manufacturing and exploit the associated benefits. We also provided formulations to minimize cycle time and pressure drop of the melt simultaneously in the modified design using integrated multi-objective optimization and multidisciplinary design optimization framework. The proposed flexible multi-cavity-inserts are designed to mold multiple part family members whose geometric dimensions are allowed to vary within a permissible limit. Hence, there exists uncertainty in the input variables. Two approaches: robustness-based design optimization and reliability-based design optimization are used to handle the uncertainty in the input variables. A case study is presented to numerically illustrate the implementation of the proposed optimization frameworks. Our formulations provide Pareto optimal design options for the flexible multi-cavity-insert that will open opportunities to produce small plastic parts having dimensional similarity economically.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability

MATLAB R2018a is used to solve the RDO/AAO and RBDO/AAO formulations presented in Figs. 9 and 10, respectively. The MATLAB codes could be made available on request by emailing the corresponding author.

Abbreviations

\(A_{j}\) :

System compatibility equation corresponding to jth discipline

\(A_{part}\) :

Projected area of the part to be molded on the parting plane, m2

\(A_{proj}\) :

Total projected area of the cavity (along-with runner system), m2

\(c_{mc}\) :

Cost of machining master cavity in the injection mold insert

\(c_{mm}\) :

Cost of mold material per kg

\(c_{r}\) :

Cost of machining runners in the injection mold insert

\({\varvec{cv}}\) :

Vector of all disciplinary coupling variables

\(d_{p}\) :

Depth of the part, m

\({\varvec{d}}\) :

Vector of deterministic design variables (\({\varvec{x}}_{d}\)) as well as the mean values of the random design variables (\({\varvec{x}}_{r}\)), i.e., \({\varvec{d}} = \left[ {{\varvec{x}}_{d} ,{\varvec{\mu}}_{{x_{r} }} } \right]\)

\(d_{gate}\) :

Diameter of gate, m

\(d_{release}\) :

Mold opening distance, m

\(d_{runner}\) :

Diameter of runner, m

\(d_{sprue}\) :

Initial diameter of sprue, m

\(\overline{d}_{sprue}\) :

Mean diameter of sprue, m

\(E(g_{j} )\) :

Expected value or mean of the jth inequality constraint

\(f\) :

Objective function

\(F_{clamp\;max}\) :

Maximum clamping force, N

\(F_{{{\varvec{cv}}_{j} }}\) :

Functional relationship representing jth coupling variable

\(g\) :

Inequality constraint

\(h\) :

Equality constraint

\(k_{v}\) :

Viscosity evaluated at a shear rate of one reciprocal second, Pa sn

\(l_{gate}\) :

Length of gate, m

\(l_{part}\) :

Length of the part, m

\(l_{runner}\) :

Length of runner, m

\(l_{sprue}\) :

Length of sprue, m

\({\varvec{lb}}\) :

Vector of lower bounds of design variables

\({\varvec{LB}}\) :

Vector of lower bounds of inequality constraints

\(Max_{D}\) :

Maximum dimension corresponding to the largest cross-section, perpendicular to the direction of melt flow, m

\(min_{D}\) :

Minimum dimension corresponding to the smallest cross-section, perpendicular to the direction of melt flow, m

\(n\) :

Power law index

\(n_{c}\) :

Number of part cavities

\(n_{pfm}\) :

Number of part-family members

\(n_{rs}\) :

Number of ramification streams

\(ncv\) :

Number of the coupling variables

\(nddv\) :

Numbers of the deterministic design variables

\(nrdv\) :

Numbers of the random design variables

\({\varvec{p}}\) :

Vector of random design parameters

\(P_{{f_{j} }}\) :

Failure probability for jth inequality constraint

\(P_{{f_{j} }}^{allowable}\) :

Allowable (maximum) failure probability for jth inequality constraint

\(P_{inj}\) :

Required injection power, W

\(Q\) :

Volumetric flow rate of the melt, m3/s

\(Q_{max}\) :

Maximum volumetric flow rate of the melt, m3/s

\({\varvec{r}}\) :

Vector of all the random variables, i.e., \({\varvec{r}} = \left[ {{\varvec{x}}_{r} ,\user2{ p}} \right]\)

\(T_{eject}\) :

Recommended part ejection temperature, °C

\(t_{inj}\) :

Injection time, s

\(T_{melt}\) :

Recommended polymer melt temperature, °C

\(T_{mold}\) :

Recommended mold temperature, °C

\({\varvec{u}}\) :

Standard normal space vector corresponding to the vector of random variables, \({\varvec{r}}\)

\(\Vert{\varvec{u}}\Vert\) :

Euclidean norm of vector \({\varvec{u}}\)

\({\varvec{u}}^{*}\) :

Most probable point (MPP) of failure

\({\varvec{ub}}\) :

Vector of upper bounds of design variables

\({\varvec{UB}}\) :

Vector of upper bounds of inequality constraints

\(\overline{v}_{F}\) :

Velocity of the flow front, m/s

\(v_{i}\) :

Volume of injection mold insert, m3

\(v_{p}\) :

Volume of the injection molded part, m3

\(v_{si}\) :

Volume of injection mold sub-insert, m3

\({\varvec{x}}_{d}\) :

Vector of deterministic design variables

\({\varvec{x}}_{r}\) :

Vector of random design variables

\(\alpha\) :

Thermal diffusivity of the polymer material, m2/s

\(\alpha_{sprue}\) :

Draft angle of sprue, degree

\(\beta\) :

Reliability index

\(\beta^{T} { }\) :

Target reliability index

\(\dot{\gamma }_{max}\) :

Maximum shear rate for the plastic, s–1

\(\varepsilon\) :

Percentage shrinkage rate of the polymer from injection temperature to room temperature

\(\eta_{aeff}\) :

Apparent effective viscosity, Pa s

\({\varvec{\mu}}_{{x_{r} }}\) :

Vector of mean values of the random design variables

\(\mu_{f} { }\) :

Mean value of the objective function

\({\varvec{\mu}}_{p}\) :

Vector of mean values of random design parameters

\(\rho_{mm}\) :

Density of injection mold material, kg/m3

\(\sigma \left( {{\varvec{x}}_{r} } \right){ }\) :

Vector of standard deviations of the random design variables

\(\sigma_{{cv_{j} }}^{2}\) :

Variance of jth coupling variable

\(\sigma_{{d_{i} }}^{2}\) :

Variance of ith random design variable

\(\sigma_{{g_{j} }}\) :

Standard deviation of the jth inequality constraint

\(\sigma_{{p_{i} }}^{2} { }\) :

Variance of ith random design parameter

\(\sigma_{f}\) :

Standard deviation of the objective function

\(\varphi\) :

Ratio between width and thickness of the mold cavity

\({\Phi }\) :

Standard Gaussian cumulative distribution function (CDF)

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Authors and Affiliations

  1. Department of Industrial and Production Engineering, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh

    Nazmul Hasan & Kais Zaman

  2. Department of Systems and Industrial Engineering, The University of Arizona, Tucson, AZ, 85721, USA

    Nazmul Hasan

  3. Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, IA, 50011, USA

    Pramiti Sarker

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  1. Nazmul Hasan
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  2. Pramiti Sarker
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Hasan, N., Sarker, P. & Zaman, K. Multidisciplinary robust and reliability-based design optimization of injection molding system. Int J Interact Des Manuf 17, 2957–2975 (2023). https://doi.org/10.1007/s12008-022-01139-x

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