There is a controverial yet simple rule: Don't discuss solution with the client, discuss requirements.
It is almost by definition that client knows much less about what it takes to build a robust solution than the craftsman, in the opposite case they would have built it themselves or hired someone less qualified and told them exactly what needs to be done.
It is unrealistic to expect that you will be able to teach the customer everything you know, and although you can be expected to justify most important decisions and trade-off you've made as part of the design the customer will need some basic domain knowledge to understand your choices.
I understand you've listened to the client carefully and this is the best solution to the problem as you understood it you could come up with given all your accumulated expertise and the timeframe. If the customer does not like the solution this is probably either because:
You didn't fully understand the problem that needs to be solved or didn’t undertake a thourough testing of the solution. Back to the drawing board then.
The initial requirements have changed. Not that unusual; make sure you’re still happy with profitability or any other interest you’ve got in the job. Study new requirements and devise a solution.
The work standard is below expected. Well, you’ve done your best, haven’t you? Subsequently you won’t learn any better overnight, especially by doing something you know is wrong.
The customer does not appreciate the standard of your work. Well, there is nothing you can do, really.
As the bottom line your customer has the right to reject a solution and during everyone's work life that is guaranteed to happen more than once no matter how good the solution is. Don’t loose your sleep over it, just move on. In the long term, this relationship won’t do you much good anyway.
On the other hand probably the next greatest thing about loving your job you can do as a professional is to develop an impartial standpoint. You won’t be able to do either if you let folks who is ignorant about the subject to mess up with your work, this is a certainty. Quite simply: “take the solution as it is or we need to discuss the original problem again".