The Relationship Between Experimental Design and Non-linearity It's well known that the classical ANOVA (and ANCOVA) experimental design techniques can be expressed as linear regression (with categorical factors). However, linear regression is just that – it enforces a linear relationship between the outcome variable. If the true relationship between the outcome variable and treatment variable is not linear, a linear regression cannot tell us this (Reference:https://www.stat.cmu.edu/~cshalizi/mreg/15/lectures/03/lecture-03.pdf)  However, it appears to me that there is nothing a priori (when designing an experiment and the data generating process is unknown) that should exclude the true relationship between the treatment and the outcome variable being non-linear. Thus, what models are available to experimentalists that attempt to discern significance and effect sizes of experimental treatments under the assumption that the relationship between treatment factors and outcome variables are nonlinear. 
 A: You say " ... that the classical ANOVA (and ANCOVA) experimental design techniques can be expressed as linear regression ... ". I think this is a confusion, fundamental experimental design concepts such as blocking, randomization, replication (and also treatment design concepts), have nothing to do with linear models in itself. This concepts are equally applicable when the analysis needs nonlinear models. It is maybe unfortunate that experimental design as a topic is confounded with some important models used for analyzing the results from the experiments. 
That being said, experimental design for non-linear models presents some new problems compared to the linear case. A presentation is this. In reality the problems mostly occur with the treatment design, that is, the choice of values for the covariables. With linear models this can be done without knowledge of the parameter values, for nonlinear models some knowledge of parameter values is indispensable. This leads to Bayesian ideas being useful.
