  In simulation-based engineering design with time-consuming simulators,
Gaussian process (GP) models are widely used as fast emulators to speed up the
design optimization process. In its most commonly used form, the input of GP is
a simple list of design parameters. With rapid development of additive
manufacturing (also known as 3D printing), design inputs with 2D/3D spatial
information become prevalent in some applications, for example, neighboring
relations between pixels/voxels and material distributions in heterogeneous
materials. Such spatial information, vital to 3D printed designs, is hard to
incorporate into existing GP models with common kernels such as squared
exponential or Mat\'ern. In this work, we propose to embed a generalized
distance measure into a GP kernel, offering a novel and convenient technique to
incorporate spatial information from freeform 3D printed designs into the GP
framework. The proposed method allows complex design problems for 3D printed
objects to take advantage of a plethora of tools available from the GP
surrogate-based simulation optimization such as designed experiments and
GP-based optimizations including Bayesian optimization. We investigate the
properties of the proposed method and illustrate its performance by several
numerical examples of 3D printed antennas. The dataset is publicly available
at: https://github.com/xichennn/GP_dataset.
